{"id":266,"date":"2025-02-13T22:45:47","date_gmt":"2025-02-13T22:45:47","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/continuity\/"},"modified":"2025-08-15T19:31:16","modified_gmt":"2025-08-15T19:31:16","slug":"continuity","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/continuity\/","title":{"raw":"Continuity: Learn It 1","rendered":"Continuity: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Determine whether a function is continuous at a number.<\/li>\r\n \t<li style=\"font-weight: 400;\">Determine the input values for which a function is discontinuous.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">A function that has no holes or breaks in its graph is known as a <strong>continuous function<\/strong>. Temperature as a function of time is an example of a continuous function.\r\n\r\n<\/section>If temperature represents a continuous function, what kind of function would not be continuous? Consider an example of dollars expressed as a function of hours of parking. Let\u2019s create the function [latex]D[\/latex], where [latex]D\\left(x\\right)[\/latex] is the output representing cost in dollars for parking [latex]x[\/latex] number of hours.\r\n\r\nSuppose a parking garage charges $4.00 per hour or fraction of an hour, with a $24 per day maximum charge. Park for two hours and five minutes and the charge is $12. Park an additional hour and the charge is $16. We can never be charged $13, $14, or $15. There are real numbers between 12 and 16 that the function never outputs. There are breaks in the function\u2019s graph for this 24-hour period, points at which the price of parking jumps instantaneously by several dollars.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"977\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185310\/CNX_Precalc_Figure_12_03_0022.jpg\" alt=\"Graph of function that maps the time since midnight to the temperature. The x-axis represents the hours parked from 0 to 24. The y-axis represents dollars amounting from 0 to 28. The function is a step-function.\" width=\"977\" height=\"361\" \/> Parking-garage charges form a discontinuous function.[\/caption]\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>stepwise function<\/h3>\r\nA function that remains level for an interval and then jumps instantaneously to a higher value is called a <strong>stepwise function<\/strong>. This function is an example.\r\n\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>discontinuous function<\/h3>\r\nA function that has any hole or break in its graph is known as a <strong>discontinuous function<\/strong>.\r\n\r\n<\/section>So how can we decide if a function is continuous at a particular number? We can check three different conditions.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Let\u2019s use the function [latex]y=f\\left(x\\right)[\/latex] shown below as an example.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185313\/CNX_Precalc_Figure_12_03_0032.jpg\" alt=\"Graph of an increasing function with a discontinuity at (a, f(a)).\" width=\"487\" height=\"251\" \/>\r\n\r\n<strong>Condition 1<\/strong> According to Condition 1, the function [latex]f\\left(a\\right)[\/latex] defined at [latex]x=a[\/latex] must exist. In other words, there is a <em>y<\/em>-coordinate at [latex]x=a[\/latex].\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185315\/CNX_Precalc_Figure_12_03_0042.jpg\" alt=\"Graph of an increasing function with a discontinuity at (a, 2). The point (a, f(a)) is directly below the hole.\" width=\"487\" height=\"251\" \/>\r\n\r\n<strong>Condition 2<\/strong> According to Condition 2, at [latex]x=a[\/latex] the limit, written [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)[\/latex], must exist. This means that at [latex]x=a[\/latex] the left-hand limit must equal the right-hand limit. Notice as the graph of [latex]f[\/latex] approaches [latex]x=a[\/latex] from the left and right, the same <em>y<\/em>-coordinate is approached. Therefore, Condition 2 is satisfied. However, there could still be a hole in the graph at [latex]x=a[\/latex] .\r\n\r\n<strong>Condition 3<\/strong> According to Condition 3, the corresponding [latex]y[\/latex] coordinate at [latex]x=a[\/latex] fills in the hole in the graph of [latex]f[\/latex]. This is written [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=f\\left(a\\right)[\/latex].\r\n\r\nSatisfying all three conditions means that the function is continuous. All three conditions are satisfied for the function represented below so the function is continuous as [latex]x=a[\/latex].\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\/3675\/2018\/09\/27185319\/CNX_Precalc_Figure_12_03_0052.jpg\" alt=\"Graph of an increasing function with filled-in discontinuity at (a, f(a)).\" width=\"487\" height=\"251\" \/> All three conditions are satisfied. The function is continuous at [latex]x=a[\/latex] .[\/caption]Below are several examples of graphs of functions that are not continuous at [latex]x=a[\/latex] and the condition or conditions that fail.\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\/3675\/2018\/09\/27185322\/CNX_Precalc_Figure_12_03_0062.jpg\" alt=\"Graph of an increasing function with a discontinuity at (a, f(a)).\" width=\"487\" height=\"251\" \/> \u00a0Condition 2 is satisfied. Conditions 1 and 3 both fail.[\/caption]\r\n\r\n&nbsp;\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\/3675\/2018\/09\/27185324\/CNX_Precalc_Figure_12_03_0072.jpg\" alt=\"Graph of an increasing function with a discontinuity at (a, 2). The point (a, f(a)) is directly below the hole.\" width=\"487\" height=\"256\" \/> Conditions 1 and 2 are both satisfied. Condition 3 fails.[\/caption]\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\/3675\/2018\/09\/27185327\/CNX_Precalc_Figure_12_03_0082.jpg\" alt=\"Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)), which is closed, and another increasing segment from (a, f(a)-1), which is open, to positive infinity.\" width=\"487\" height=\"250\" \/> Condition 1 is satisfied. Conditions 2 and 3 fail.[\/caption]\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\/3675\/2018\/09\/27185329\/CNX_Precalc_Figure_12_03_0092.jpg\" alt=\"Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)) and another increasing segment from (a, f(a) - 1) to positive infinity. This graph does not include the point (a, f(a)).\" width=\"487\" height=\"251\" \/> Conditions 1, 2, and 3 all fail.[\/caption]\r\n\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>definition of continuity<\/h3>\r\nA function [latex]f\\left(x\\right)[\/latex] is <strong>continuous<\/strong> at [latex]x=a[\/latex] provided all three of the following conditions hold true:\r\n\r\nCondition 1: [latex]f(a)[\/latex] exists.\r\n\r\nCondition 2: [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)[\/latex] exists at [latex]x=a[\/latex].\r\n\r\nCondition 2: [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)=f(a)[\/latex].\r\n\r\nIf a function [latex]f\\left(x\\right)[\/latex] is not continuous at [latex]x=a[\/latex], the function is <strong>discontinuous<\/strong> at [latex]x=a[\/latex] .\r\n\r\n<\/section>\r\n<dl id=\"fs-id1165135700059\" class=\"definition\">\r\n \t<dd id=\"fs-id1165135700065\"><\/dd>\r\n<\/dl>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li style=\"font-weight: 400;\">Determine whether a function is continuous at a number.<\/li>\n<li style=\"font-weight: 400;\">Determine the input values for which a function is discontinuous.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">A function that has no holes or breaks in its graph is known as a <strong>continuous function<\/strong>. Temperature as a function of time is an example of a continuous function.<\/p>\n<\/section>\n<p>If temperature represents a continuous function, what kind of function would not be continuous? Consider an example of dollars expressed as a function of hours of parking. Let\u2019s create the function [latex]D[\/latex], where [latex]D\\left(x\\right)[\/latex] is the output representing cost in dollars for parking [latex]x[\/latex] number of hours.<\/p>\n<p>Suppose a parking garage charges $4.00 per hour or fraction of an hour, with a $24 per day maximum charge. Park for two hours and five minutes and the charge is $12. Park an additional hour and the charge is $16. We can never be charged $13, $14, or $15. There are real numbers between 12 and 16 that the function never outputs. There are breaks in the function\u2019s graph for this 24-hour period, points at which the price of parking jumps instantaneously by several dollars.<\/p>\n<figure style=\"width: 977px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185310\/CNX_Precalc_Figure_12_03_0022.jpg\" alt=\"Graph of function that maps the time since midnight to the temperature. The x-axis represents the hours parked from 0 to 24. The y-axis represents dollars amounting from 0 to 28. The function is a step-function.\" width=\"977\" height=\"361\" \/><figcaption class=\"wp-caption-text\">Parking-garage charges form a discontinuous function.<\/figcaption><\/figure>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>stepwise function<\/h3>\n<p>A function that remains level for an interval and then jumps instantaneously to a higher value is called a <strong>stepwise function<\/strong>. This function is an example.<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>discontinuous function<\/h3>\n<p>A function that has any hole or break in its graph is known as a <strong>discontinuous function<\/strong>.<\/p>\n<\/section>\n<p>So how can we decide if a function is continuous at a particular number? We can check three different conditions.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Let\u2019s use the function [latex]y=f\\left(x\\right)[\/latex] shown below as an example.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185313\/CNX_Precalc_Figure_12_03_0032.jpg\" alt=\"Graph of an increasing function with a discontinuity at (a, f(a)).\" width=\"487\" height=\"251\" \/><\/p>\n<p><strong>Condition 1<\/strong> According to Condition 1, the function [latex]f\\left(a\\right)[\/latex] defined at [latex]x=a[\/latex] must exist. In other words, there is a <em>y<\/em>-coordinate at [latex]x=a[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185315\/CNX_Precalc_Figure_12_03_0042.jpg\" alt=\"Graph of an increasing function with a discontinuity at (a, 2). The point (a, f(a)) is directly below the hole.\" width=\"487\" height=\"251\" \/><\/p>\n<p><strong>Condition 2<\/strong> According to Condition 2, at [latex]x=a[\/latex] the limit, written [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)[\/latex], must exist. This means that at [latex]x=a[\/latex] the left-hand limit must equal the right-hand limit. Notice as the graph of [latex]f[\/latex] approaches [latex]x=a[\/latex] from the left and right, the same <em>y<\/em>-coordinate is approached. Therefore, Condition 2 is satisfied. However, there could still be a hole in the graph at [latex]x=a[\/latex] .<\/p>\n<p><strong>Condition 3<\/strong> According to Condition 3, the corresponding [latex]y[\/latex] coordinate at [latex]x=a[\/latex] fills in the hole in the graph of [latex]f[\/latex]. This is written [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=f\\left(a\\right)[\/latex].<\/p>\n<p>Satisfying all three conditions means that the function is continuous. All three conditions are satisfied for the function represented below so the function is continuous as [latex]x=a[\/latex].<\/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\/3675\/2018\/09\/27185319\/CNX_Precalc_Figure_12_03_0052.jpg\" alt=\"Graph of an increasing function with filled-in discontinuity at (a, f(a)).\" width=\"487\" height=\"251\" \/><figcaption class=\"wp-caption-text\">All three conditions are satisfied. The function is continuous at [latex]x=a[\/latex] .<\/figcaption><\/figure>\n<p>Below are several examples of graphs of functions that are not continuous at [latex]x=a[\/latex] and the condition or conditions that fail.<\/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\/3675\/2018\/09\/27185322\/CNX_Precalc_Figure_12_03_0062.jpg\" alt=\"Graph of an increasing function with a discontinuity at (a, f(a)).\" width=\"487\" height=\"251\" \/><figcaption class=\"wp-caption-text\">\u00a0Condition 2 is satisfied. Conditions 1 and 3 both fail.<\/figcaption><\/figure>\n<p>&nbsp;<\/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\/3675\/2018\/09\/27185324\/CNX_Precalc_Figure_12_03_0072.jpg\" alt=\"Graph of an increasing function with a discontinuity at (a, 2). The point (a, f(a)) is directly below the hole.\" width=\"487\" height=\"256\" \/><figcaption class=\"wp-caption-text\">Conditions 1 and 2 are both satisfied. Condition 3 fails.<\/figcaption><\/figure>\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\/3675\/2018\/09\/27185327\/CNX_Precalc_Figure_12_03_0082.jpg\" alt=\"Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)), which is closed, and another increasing segment from (a, f(a)-1), which is open, to positive infinity.\" width=\"487\" height=\"250\" \/><figcaption class=\"wp-caption-text\">Condition 1 is satisfied. Conditions 2 and 3 fail.<\/figcaption><\/figure>\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\/3675\/2018\/09\/27185329\/CNX_Precalc_Figure_12_03_0092.jpg\" alt=\"Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)) and another increasing segment from (a, f(a) - 1) to positive infinity. This graph does not include the point (a, f(a)).\" width=\"487\" height=\"251\" \/><figcaption class=\"wp-caption-text\">Conditions 1, 2, and 3 all fail.<\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>definition of continuity<\/h3>\n<p>A function [latex]f\\left(x\\right)[\/latex] is <strong>continuous<\/strong> at [latex]x=a[\/latex] provided all three of the following conditions hold true:<\/p>\n<p>Condition 1: [latex]f(a)[\/latex] exists.<\/p>\n<p>Condition 2: [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)[\/latex] exists at [latex]x=a[\/latex].<\/p>\n<p>Condition 2: [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)=f(a)[\/latex].<\/p>\n<p>If a function [latex]f\\left(x\\right)[\/latex] is not continuous at [latex]x=a[\/latex], the function is <strong>discontinuous<\/strong> at [latex]x=a[\/latex] .<\/p>\n<\/section>\n<dl id=\"fs-id1165135700059\" class=\"definition\">\n<dd id=\"fs-id1165135700065\"><\/dd>\n<\/dl>\n","protected":false},"author":6,"menu_order":18,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"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":263,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/266"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/266\/revisions"}],"predecessor-version":[{"id":2999,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/266\/revisions\/2999"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/263"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/266\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=266"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=266"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=266"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=266"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}