{"id":202,"date":"2023-09-20T22:48:14","date_gmt":"2023-09-20T22:48:14","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/continuity-at-a-point\/"},"modified":"2025-08-17T22:25:51","modified_gmt":"2025-08-17T22:25:51","slug":"continuity-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/continuity-learn-it-1\/","title":{"raw":"Continuity: Learn It 1","rendered":"Continuity: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Outline the three criteria a function must meet to be continuous at a specific point<\/li>\r\n\t<li>Explain the different types of breaks a function can have that make it not continuous<\/li>\r\n\t<li>Explain what it means for a function to be continuous over a range of values<\/li>\r\n\t<li>Explain the rule for calculating limits of functions that are combined<\/li>\r\n\t<li>Show how a continuous function reaches every value between its start and end points using the Intermediate Value Theorem<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Continuity<\/h2>\r\n<p>Many functions can be traced without lifting your pencil, indicating they are continuous. Functions that cannot be traced this way have points of discontinuity. To understand continuity at a point, consider the following conditions:<\/p>\r\n<ol>\r\n\t<li>\r\n<p><strong>The function [latex]f(a)[\/latex] is defined<\/strong>:<\/p>\r\n<ul>\r\n\t<li>The function must have a value at [latex]a[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>\r\n<p><strong>The limit [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] exists<\/strong>:<\/p>\r\n<ul>\r\n\t<li>The value that [latex]f(x)[\/latex] approaches as [latex]x[\/latex] gets closer to [latex]a[\/latex] must exist.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>\r\n<p><strong>The limit equals the function value [latex]\\underset{x\\to a}{\\lim}f(x)=f(a)[\/latex]<\/strong>:<\/p>\r\n<ul>\r\n\t<li>The value that [latex]f(x)[\/latex] approaches must be the same as the value of the function at [latex]a[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<section class=\"textbox example\">\r\n<p>Let's illustrate these conditions:<br \/>\r\n<br \/>\r\n<\/p>\r\n<ul>\r\n\t<li>\r\n<p><strong>Undefined Function<\/strong>:<\/p>\r\n<ul>\r\n\t<li>If [latex]f(a)[\/latex] is not defined, the function is not continuous at [latex]a[\/latex].<br \/>\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203502\/CNX_Calc_Figure_02_04_001.jpg\" alt=\"A graph of an increasing linear function f(x) which crosses the x axis from quadrant three to quadrant two and which crosses the y axis from quadrant two to quadrant one. A point a greater than zero is marked on the x axis. The point on the function f(x) above a is an open circle; the function is not defined at a.\" width=\"325\" height=\"277\" \/> Figure 1. The function [latex]f(x)[\/latex] is not continuous at a because [latex]f(a)[\/latex] is undefined.[\/caption]\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>\r\n<p><strong>Non-existent Limit<\/strong>:<\/p>\r\n<ul>\r\n\t<li>Even if [latex]f(a)[\/latex] is defined, if [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] does not exist, the function is not continuous at [latex]a[\/latex].<br \/>\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203504\/CNX_Calc_Figure_02_04_002.jpg\" alt=\"The graph of a piecewise function f(x) with two parts. The first part is an increasing linear function that crosses from quadrant three to quadrant one at the origin. A point a greater than zero is marked on the x axis. At fa. on this segment, there is a solid circle. The other segment is also an increasing linear function. It exists in quadrant one for values of x greater than a. At x=a, this segment has an open circle.\" width=\"325\" height=\"277\" \/> Figure 2. The function [latex]f(x)[\/latex] is not continuous at a because [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] does not exist.[\/caption]\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>\r\n<p><strong>Limit Not Equal to Function Value<\/strong>:<\/p>\r\n<ul>\r\n\t<li>If [latex]f(a)[\/latex] is defined and [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] exists, but they are not equal, the function is not continuous at [latex]a[\/latex].<br \/>\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203506\/CNX_Calc_Figure_02_04_003.jpg\" alt=\"The graph of a piecewise function with two parts. The first part is an increasing linear function that crosses the x axis from quadrant three to quadrant two and which crosses the y axis from quadrant two to quadrant one. A point a greater than zero is marked on the x axis. At this point, there is an open circle on the linear function. The second part is a point at x=a above the line.\" width=\"325\" height=\"277\" \/> Figure 3. The function [latex]f(x)[\/latex] is not continuous at a because [latex]\\underset{x\\to a}{\\lim}f(x)\\ne f(a)[\/latex].[\/caption]\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3><strong>continuity <\/strong><\/h3>\r\n<p id=\"fs-id1170573381317\">A function [latex]f(x)[\/latex] is <strong>continuous at a point<\/strong> [latex]a[\/latex] if and only if the following three conditions are satisfied:<\/p>\r\n<ol id=\"fs-id1170571094915\">\r\n\t<li>[latex]f(a)[\/latex] is defined<\/li>\r\n\t<li>[latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] exists<\/li>\r\n\t<li>[latex]\\underset{x\\to a}{\\lim}f(x)=f(a)[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170573370662\">A function is <strong>discontinuous at a point<\/strong> [latex]a[\/latex] if it fails to be continuous at [latex]a[\/latex].<\/p>\r\n<\/section>\r\n<p id=\"fs-id1170573281579\">The following procedure can be used to analyze the continuity of a function at a point using this definition.<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How to: Determine Continuity at a Point<\/strong><\/p>\r\n<ol id=\"fs-id1170571103215\">\r\n\t<li>Check to see if [latex]f(a)[\/latex] is defined.\r\n\r\n<ul>\r\n\t<li>If [latex]f(a)[\/latex] is undefined, we need go no further. The function is not continuous at [latex]a[\/latex].<\/li>\r\n\t<li>If [latex]f(a)[\/latex] is defined, continue to step 2.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Compute [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex]. In some cases, we may need to do this by first computing [latex]\\underset{x\\to a^-}{\\lim}f(x)[\/latex] and [latex]\\underset{x\\to a^+}{\\lim}f(x)[\/latex].\r\n\r\n<ul>\r\n\t<li>If [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] does not exist (that is, it is not a real number), then the function is not continuous at [latex]a[\/latex].<\/li>\r\n\t<li>If [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] exists, then continue to step 3.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Compare [latex]f(a)[\/latex] and [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex].\r\n\r\n<ul>\r\n\t<li>If [latex]\\underset{x\\to a}{\\lim}f(x)\\ne f(a)[\/latex], then the function is not continuous at [latex]a[\/latex].<\/li>\r\n\t<li>If [latex]\\underset{x\\to a}{\\lim}f(x)=f(a)[\/latex], then the function is continuous at [latex]a[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/section>\r\n<p id=\"fs-id1170573570770\">The next three examples demonstrate how to apply this definition to determine whether a function is continuous at a given point. These examples illustrate situations in which each of the conditions for continuity in the definition succeed or fail.<\/p>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170573258704\">Using the definition, determine whether the function [latex]f(x)=\\dfrac{(x^2-4)}{(x-2)}[\/latex] is continuous at [latex]x=2[\/latex]. Justify the conclusion.<\/p>\r\n<p>[reveal-answer q=\"fs-id1170573331408\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170573331408\"]<\/p>\r\n<p id=\"fs-id1170573331408\">Let\u2019s begin by trying to calculate [latex]f(2)[\/latex]. We can see that [latex]f(2)=\\frac{0}{0}[\/latex], which is undefined. Therefore, [latex]f(x)=\\frac{x^2-4}{x-2}[\/latex] is discontinuous at 2 because [latex]f(2)[\/latex] is undefined. The graph of [latex]f(x)[\/latex] is shown in Figure 4.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203508\/CNX_Calc_Figure_02_04_004.jpg\" alt=\"A graph of the given function. There is a line which crosses the x axis from quadrant three to quadrant two and which crosses the y axis from quadrant two to quadrant one. At a point in quadrant one, there is an open circle where the function is not defined.\" width=\"487\" height=\"425\" \/> Figure 4. The function [latex]f(x)[\/latex] is discontinuous at 2 because [latex]f(2)[\/latex] is undefined.[\/caption]\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170573370721\">Using the definition, determine whether the function [latex]f(x)=\\begin{cases} -x^2+4 &amp; \\text{ if } \\, x \\le 3 \\\\ 4x-8 &amp; \\text{ if } \\, x &gt; 3 \\end{cases}[\/latex] is continuous at [latex]x=3[\/latex]. Justify the conclusion.<\/p>\r\n<p>[reveal-answer q=\"fs-id1170573349668\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170573349668\"]<\/p>\r\n<p id=\"fs-id1170573349668\">Let\u2019s begin by trying to calculate [latex]f(3)[\/latex].<\/p>\r\n<div id=\"fs-id1170573405058\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(3)=-(3^2)+4=-5[\/latex]<\/div>\r\n<p id=\"fs-id1170573307347\">Thus, [latex]f(3)[\/latex] is defined. Next, we calculate [latex]\\underset{x\\to 3}{\\lim}f(x)[\/latex]. To do this, we must compute [latex]\\underset{x\\to 3^-}{\\lim}f(x)[\/latex] and [latex]\\underset{x\\to 3^+}{\\lim}f(x)[\/latex]:<\/p>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 3^-}{\\lim}f(x)=-(3^2)+4=-5[\/latex]<\/div>\r\n<p id=\"fs-id1170573400834\">and<\/p>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 3^+}{\\lim}f(x)=4(3)-8=4[\/latex]<\/div>\r\n<p id=\"fs-id1170573414763\">Therefore, [latex]\\underset{x\\to 3}{\\lim}f(x)[\/latex] does not exist. Thus, [latex]f(x)[\/latex] is not continuous at [latex]3[\/latex]. The graph of [latex]f(x)[\/latex] is shown in Figure 5.<\/p>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203511\/CNX_Calc_Figure_02_04_005.jpg\" alt=\"&quot;A\" width=\"487\" height=\"575\" \/> Graph showing the discontinuous function f(x)[\/caption]\r\n\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170573502415\">Using the definition, determine whether the function [latex]f(x)=\\begin{cases} \\frac{\\sin x}{x} &amp; \\text{ if } \\, x \\ne 0 \\\\ 1 &amp; \\text{ if } \\, x = 0 \\end{cases}[\/latex] is continuous at [latex]x=0[\/latex].<\/p>\r\n<p>[reveal-answer q=\"fs-id1170573382461\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170573382461\"]<\/p>\r\n<p id=\"fs-id1170573382461\">First, observe that<\/p>\r\n<div id=\"fs-id1170573569453\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(0)=1[\/latex]<\/div>\r\n<p id=\"fs-id1170573367184\">Next,<\/p>\r\n<div id=\"fs-id1170573442773\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 0}{\\lim}f(x)=\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{x}=1[\/latex]<\/div>\r\n<p id=\"fs-id1170573394612\">Last, compare [latex]f(0)[\/latex] and [latex]\\underset{x\\to 1}{\\lim}f(x)[\/latex]. We see that<\/p>\r\n<div id=\"fs-id1170573332534\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(0)=1=\\underset{x\\to 0}{\\lim}f(x)[\/latex]<\/div>\r\n<p id=\"fs-id1170573394599\">Since all three of the conditions in the definition of continuity are satisfied, [latex]f(x)[\/latex] is continuous at [latex]x=0[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\">\r\n<p>[caption]Watch the following video to see the worked solutions to the three previous examples. [\/caption]<\/p>\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/BXUu5bG1CXU?controls=0&amp;start=182&amp;end=412&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"266833\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.4Continuity182to412_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"2.4 Continuity\" here (opens in new window).<\/a>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]288279[\/ohm_question]<\/p>\r\n<\/section>\r\n<p id=\"fs-id1170573513079\">By applying the definition of continuity and previously established theorems concerning the evaluation of limits, we can state the following theorem.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>continuity of polynomials and rational functions<\/h3>\r\n<p>Polynomials and rational functions are continuous at every point in their domains.<\/p>\r\n<\/section>\r\n<section class=\"textbox connectIt\">\r\n<p style=\"text-align: center;\"><strong>Proof<\/strong><\/p>\r\n<hr \/>\r\n<p id=\"fs-id1170573326736\">Previously, we showed that if [latex]p(x)[\/latex] and [latex]q(x)[\/latex] are polynomials, [latex]\\underset{x\\to a}{\\lim}p(x)=p(a)[\/latex] for every polynomial [latex]p(x)[\/latex] and [latex]\\underset{x\\to a}{\\lim}\\dfrac{p(x)}{q(x)}=\\dfrac{p(a)}{q(a)}[\/latex] as long as [latex]q(a)\\ne 0[\/latex]. Therefore, polynomials and rational functions are continuous on their domains.<\/p>\r\n<p>[latex]_\\blacksquare[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox recall\">\r\n<p>The domain of every polynomial function is <strong>all real numbers.\u00a0<\/strong><\/p>\r\n<p>The domain of a rational functional can be found by:<\/p>\r\n<ol>\r\n\t<li>Set the denominator equal to zero.<\/li>\r\n\t<li>Solve to find the values of the variable that cause the denominator to equal zero.<\/li>\r\n\t<li>The domain contains all real numbers except those found in Step 2.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<p id=\"fs-id1170573335424\">We now apply continuity of polynomials and rational functions to determine the points at which a given rational function is continuous.<\/p>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170573365757\">For what values of [latex]x[\/latex] is [latex]f(x)=\\dfrac{x+1}{x-5}[\/latex] continuous?<\/p>\r\n<p>[reveal-answer q=\"fs-id1170573397031\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170573397031\"]<\/p>\r\n<p id=\"fs-id1170573397031\">The rational function [latex]f(x)=\\frac{x+1}{x-5}[\/latex] is continuous for every value of [latex]x[\/latex] except [latex]x=5[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Outline the three criteria a function must meet to be continuous at a specific point<\/li>\n<li>Explain the different types of breaks a function can have that make it not continuous<\/li>\n<li>Explain what it means for a function to be continuous over a range of values<\/li>\n<li>Explain the rule for calculating limits of functions that are combined<\/li>\n<li>Show how a continuous function reaches every value between its start and end points using the Intermediate Value Theorem<\/li>\n<\/ul>\n<\/section>\n<h2>Continuity<\/h2>\n<p>Many functions can be traced without lifting your pencil, indicating they are continuous. Functions that cannot be traced this way have points of discontinuity. To understand continuity at a point, consider the following conditions:<\/p>\n<ol>\n<li>\n<p><strong>The function [latex]f(a)[\/latex] is defined<\/strong>:<\/p>\n<ul>\n<li>The function must have a value at [latex]a[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>\n<p><strong>The limit [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] exists<\/strong>:<\/p>\n<ul>\n<li>The value that [latex]f(x)[\/latex] approaches as [latex]x[\/latex] gets closer to [latex]a[\/latex] must exist.<\/li>\n<\/ul>\n<\/li>\n<li>\n<p><strong>The limit equals the function value [latex]\\underset{x\\to a}{\\lim}f(x)=f(a)[\/latex]<\/strong>:<\/p>\n<ul>\n<li>The value that [latex]f(x)[\/latex] approaches must be the same as the value of the function at [latex]a[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<section class=\"textbox example\">\n<p>Let&#8217;s illustrate these conditions:<\/p>\n<ul>\n<li>\n<p><strong>Undefined Function<\/strong>:<\/p>\n<ul>\n<li>If [latex]f(a)[\/latex] is not defined, the function is not continuous at [latex]a[\/latex].<br \/>\n<figure style=\"width: 325px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203502\/CNX_Calc_Figure_02_04_001.jpg\" alt=\"A graph of an increasing linear function f(x) which crosses the x axis from quadrant three to quadrant two and which crosses the y axis from quadrant two to quadrant one. A point a greater than zero is marked on the x axis. The point on the function f(x) above a is an open circle; the function is not defined at a.\" width=\"325\" height=\"277\" \/><figcaption class=\"wp-caption-text\">Figure 1. The function [latex]f(x)[\/latex] is not continuous at a because [latex]f(a)[\/latex] is undefined.<\/figcaption><\/figure>\n<\/li>\n<\/ul>\n<\/li>\n<li>\n<p><strong>Non-existent Limit<\/strong>:<\/p>\n<ul>\n<li>Even if [latex]f(a)[\/latex] is defined, if [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] does not exist, the function is not continuous at [latex]a[\/latex].<br \/>\n<figure style=\"width: 325px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203504\/CNX_Calc_Figure_02_04_002.jpg\" alt=\"The graph of a piecewise function f(x) with two parts. The first part is an increasing linear function that crosses from quadrant three to quadrant one at the origin. A point a greater than zero is marked on the x axis. At fa. on this segment, there is a solid circle. The other segment is also an increasing linear function. It exists in quadrant one for values of x greater than a. At x=a, this segment has an open circle.\" width=\"325\" height=\"277\" \/><figcaption class=\"wp-caption-text\">Figure 2. The function [latex]f(x)[\/latex] is not continuous at a because [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] does not exist.<\/figcaption><\/figure>\n<\/li>\n<\/ul>\n<\/li>\n<li>\n<p><strong>Limit Not Equal to Function Value<\/strong>:<\/p>\n<ul>\n<li>If [latex]f(a)[\/latex] is defined and [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] exists, but they are not equal, the function is not continuous at [latex]a[\/latex].<br \/>\n<figure style=\"width: 325px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203506\/CNX_Calc_Figure_02_04_003.jpg\" alt=\"The graph of a piecewise function with two parts. The first part is an increasing linear function that crosses the x axis from quadrant three to quadrant two and which crosses the y axis from quadrant two to quadrant one. A point a greater than zero is marked on the x axis. At this point, there is an open circle on the linear function. The second part is a point at x=a above the line.\" width=\"325\" height=\"277\" \/><figcaption class=\"wp-caption-text\">Figure 3. The function [latex]f(x)[\/latex] is not continuous at a because [latex]\\underset{x\\to a}{\\lim}f(x)\\ne f(a)[\/latex].<\/figcaption><\/figure>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3><strong>continuity <\/strong><\/h3>\n<p id=\"fs-id1170573381317\">A function [latex]f(x)[\/latex] is <strong>continuous at a point<\/strong> [latex]a[\/latex] if and only if the following three conditions are satisfied:<\/p>\n<ol id=\"fs-id1170571094915\">\n<li>[latex]f(a)[\/latex] is defined<\/li>\n<li>[latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] exists<\/li>\n<li>[latex]\\underset{x\\to a}{\\lim}f(x)=f(a)[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1170573370662\">A function is <strong>discontinuous at a point<\/strong> [latex]a[\/latex] if it fails to be continuous at [latex]a[\/latex].<\/p>\n<\/section>\n<p id=\"fs-id1170573281579\">The following procedure can be used to analyze the continuity of a function at a point using this definition.<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How to: Determine Continuity at a Point<\/strong><\/p>\n<ol id=\"fs-id1170571103215\">\n<li>Check to see if [latex]f(a)[\/latex] is defined.\n<ul>\n<li>If [latex]f(a)[\/latex] is undefined, we need go no further. The function is not continuous at [latex]a[\/latex].<\/li>\n<li>If [latex]f(a)[\/latex] is defined, continue to step 2.<\/li>\n<\/ul>\n<\/li>\n<li>Compute [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex]. In some cases, we may need to do this by first computing [latex]\\underset{x\\to a^-}{\\lim}f(x)[\/latex] and [latex]\\underset{x\\to a^+}{\\lim}f(x)[\/latex].\n<ul>\n<li>If [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] does not exist (that is, it is not a real number), then the function is not continuous at [latex]a[\/latex].<\/li>\n<li>If [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] exists, then continue to step 3.<\/li>\n<\/ul>\n<\/li>\n<li>Compare [latex]f(a)[\/latex] and [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex].\n<ul>\n<li>If [latex]\\underset{x\\to a}{\\lim}f(x)\\ne f(a)[\/latex], then the function is not continuous at [latex]a[\/latex].<\/li>\n<li>If [latex]\\underset{x\\to a}{\\lim}f(x)=f(a)[\/latex], then the function is continuous at [latex]a[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/section>\n<p id=\"fs-id1170573570770\">The next three examples demonstrate how to apply this definition to determine whether a function is continuous at a given point. These examples illustrate situations in which each of the conditions for continuity in the definition succeed or fail.<\/p>\n<section class=\"textbox example\">\n<p id=\"fs-id1170573258704\">Using the definition, determine whether the function [latex]f(x)=\\dfrac{(x^2-4)}{(x-2)}[\/latex] is continuous at [latex]x=2[\/latex]. Justify the conclusion.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170573331408\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170573331408\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170573331408\">Let\u2019s begin by trying to calculate [latex]f(2)[\/latex]. We can see that [latex]f(2)=\\frac{0}{0}[\/latex], which is undefined. Therefore, [latex]f(x)=\\frac{x^2-4}{x-2}[\/latex] is discontinuous at 2 because [latex]f(2)[\/latex] is undefined. The graph of [latex]f(x)[\/latex] is shown in Figure 4.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203508\/CNX_Calc_Figure_02_04_004.jpg\" alt=\"A graph of the given function. There is a line which crosses the x axis from quadrant three to quadrant two and which crosses the y axis from quadrant two to quadrant one. At a point in quadrant one, there is an open circle where the function is not defined.\" width=\"487\" height=\"425\" \/><figcaption class=\"wp-caption-text\">Figure 4. The function [latex]f(x)[\/latex] is discontinuous at 2 because [latex]f(2)[\/latex] is undefined.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170573370721\">Using the definition, determine whether the function [latex]f(x)=\\begin{cases} -x^2+4 & \\text{ if } \\, x \\le 3 \\\\ 4x-8 & \\text{ if } \\, x > 3 \\end{cases}[\/latex] is continuous at [latex]x=3[\/latex]. Justify the conclusion.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170573349668\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170573349668\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170573349668\">Let\u2019s begin by trying to calculate [latex]f(3)[\/latex].<\/p>\n<div id=\"fs-id1170573405058\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(3)=-(3^2)+4=-5[\/latex]<\/div>\n<p id=\"fs-id1170573307347\">Thus, [latex]f(3)[\/latex] is defined. Next, we calculate [latex]\\underset{x\\to 3}{\\lim}f(x)[\/latex]. To do this, we must compute [latex]\\underset{x\\to 3^-}{\\lim}f(x)[\/latex] and [latex]\\underset{x\\to 3^+}{\\lim}f(x)[\/latex]:<\/p>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 3^-}{\\lim}f(x)=-(3^2)+4=-5[\/latex]<\/div>\n<p id=\"fs-id1170573400834\">and<\/p>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 3^+}{\\lim}f(x)=4(3)-8=4[\/latex]<\/div>\n<p id=\"fs-id1170573414763\">Therefore, [latex]\\underset{x\\to 3}{\\lim}f(x)[\/latex] does not exist. Thus, [latex]f(x)[\/latex] is not continuous at [latex]3[\/latex]. The graph of [latex]f(x)[\/latex] is shown in Figure 5.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203511\/CNX_Calc_Figure_02_04_005.jpg\" alt=\"&quot;A\" width=\"487\" height=\"575\" \/><figcaption class=\"wp-caption-text\">Graph showing the discontinuous function f(x)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170573502415\">Using the definition, determine whether the function [latex]f(x)=\\begin{cases} \\frac{\\sin x}{x} & \\text{ if } \\, x \\ne 0 \\\\ 1 & \\text{ if } \\, x = 0 \\end{cases}[\/latex] is continuous at [latex]x=0[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170573382461\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170573382461\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170573382461\">First, observe that<\/p>\n<div id=\"fs-id1170573569453\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(0)=1[\/latex]<\/div>\n<p id=\"fs-id1170573367184\">Next,<\/p>\n<div id=\"fs-id1170573442773\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 0}{\\lim}f(x)=\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{x}=1[\/latex]<\/div>\n<p id=\"fs-id1170573394612\">Last, compare [latex]f(0)[\/latex] and [latex]\\underset{x\\to 1}{\\lim}f(x)[\/latex]. We see that<\/p>\n<div id=\"fs-id1170573332534\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(0)=1=\\underset{x\\to 0}{\\lim}f(x)[\/latex]<\/div>\n<p id=\"fs-id1170573394599\">Since all three of the conditions in the definition of continuity are satisfied, [latex]f(x)[\/latex] is continuous at [latex]x=0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\">\nWatch the following video to see the worked solutions to the three previous examples. <\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/BXUu5bG1CXU?controls=0&amp;start=182&amp;end=412&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/button><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.4Continuity182to412_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;2.4 Continuity&#8221; here (opens in new window).<\/a><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm288279\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288279&theme=lumen&iframe_resize_id=ohm288279&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<p id=\"fs-id1170573513079\">By applying the definition of continuity and previously established theorems concerning the evaluation of limits, we can state the following theorem.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>continuity of polynomials and rational functions<\/h3>\n<p>Polynomials and rational functions are continuous at every point in their domains.<\/p>\n<\/section>\n<section class=\"textbox connectIt\">\n<p style=\"text-align: center;\"><strong>Proof<\/strong><\/p>\n<hr \/>\n<p id=\"fs-id1170573326736\">Previously, we showed that if [latex]p(x)[\/latex] and [latex]q(x)[\/latex] are polynomials, [latex]\\underset{x\\to a}{\\lim}p(x)=p(a)[\/latex] for every polynomial [latex]p(x)[\/latex] and [latex]\\underset{x\\to a}{\\lim}\\dfrac{p(x)}{q(x)}=\\dfrac{p(a)}{q(a)}[\/latex] as long as [latex]q(a)\\ne 0[\/latex]. Therefore, polynomials and rational functions are continuous on their domains.<\/p>\n<p>[latex]_\\blacksquare[\/latex]<br \/>\n<\/section>\n<section class=\"textbox recall\">\n<p>The domain of every polynomial function is <strong>all real numbers.\u00a0<\/strong><\/p>\n<p>The domain of a rational functional can be found by:<\/p>\n<ol>\n<li>Set the denominator equal to zero.<\/li>\n<li>Solve to find the values of the variable that cause the denominator to equal zero.<\/li>\n<li>The domain contains all real numbers except those found in Step 2.<\/li>\n<\/ol>\n<\/section>\n<p id=\"fs-id1170573335424\">We now apply continuity of polynomials and rational functions to determine the points at which a given rational function is continuous.<\/p>\n<section class=\"textbox example\">\n<p id=\"fs-id1170573365757\">For what values of [latex]x[\/latex] is [latex]f(x)=\\dfrac{x+1}{x-5}[\/latex] continuous?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170573397031\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170573397031\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170573397031\">The rational function [latex]f(x)=\\frac{x+1}{x-5}[\/latex] is continuous for every value of [latex]x[\/latex] except [latex]x=5[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"2.4 Continuity\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":185,"module-header":"learn_it","content_attributions":[{"type":"cc","description":"Calculus Volume 1","author":"Gilbert Strang, Edwin (Jed) Herman","organization":"OpenStax","url":"https:\/\/openstax.org\/details\/books\/calculus-volume-1","project":"","license":"cc-by-nc-sa","license_terms":"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction"},{"type":"original","description":"2.4 Continuity","author":"Ryan Melton","organization":"","url":"","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/202"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":18,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/202\/revisions"}],"predecessor-version":[{"id":4753,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/202\/revisions\/4753"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/185"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/202\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=202"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=202"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=202"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=202"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}