{"id":1613,"date":"2025-07-25T04:07:11","date_gmt":"2025-07-25T04:07:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1613"},"modified":"2026-02-13T17:46:02","modified_gmt":"2026-02-13T17:46:02","slug":"continuity-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/continuity-fresh-take\/","title":{"raw":"Continuity: Fresh Take","rendered":"Continuity: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Determine whether a function is continuous at a number.<\/li>\r\n \t<li>Determine the input values for which a function is discontinuous.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Continuity<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">A function [latex]f(x)[\/latex] is continuous at [latex]x = a[\/latex] if:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">f(a) is defined<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} f(x)[\/latex] exists<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} f(x) = f(a)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Types of Discontinuities:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Removable (hole in the graph)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Jump (function value jumps at a point)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Infinite (function approaches infinity near a point)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Continuity of Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Polynomials are continuous everywhere<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rational functions are continuous except where denominator is zero<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Analyzing Continuity:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Check if function is defined at the point<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Evaluate the limit as [latex]x[\/latex] approaches the point<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Compare the limit value with the function value at the point<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170573406319\">Using the definition, determine whether the function [latex]f(x)=\\begin{cases} 2x+1 &amp; \\text{ if } \\, x &lt; 1 \\\\ 2 &amp; \\text{ if } \\, x = 1 \\\\ -x+4 &amp; \\text{ if } \\, x &gt; 1 \\end{cases}[\/latex] is continuous at [latex]x=1[\/latex].<\/p>\r\nIf the function is not continuous at [latex]1[\/latex], indicate the condition for continuity at a point that fails to hold.\r\n\r\n[reveal-answer q=\"98945562\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"98945562\"]\r\n<p id=\"fs-id1170573418814\">Check each condition of the definition.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170573402451\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573402451\"]\r\n<p id=\"fs-id1170573402451\">[latex]f[\/latex] is not continuous at [latex]1[\/latex] because [latex]f(1)=2\\ne 3=\\underset{x\\to 1}{\\lim}f(x)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1170570999873\">For what values of [latex]x[\/latex] is [latex]f(x)=3x^4-4x^2[\/latex] continuous?<\/p>\r\n[reveal-answer q=\"fs-id1170573246209\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573246209\"]\r\n<p id=\"fs-id1170573246209\">[latex]f(x)[\/latex] is continuous at every real number.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Types of Discontinuities<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Three Main Types of Discontinuities:\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Removable (hole in the graph)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Jump (function value jumps at a point)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Infinite (function approaches infinity near a point)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Removable Discontinuity:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Limit exists at the point, but function value is different or undefined<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphically appears as a hole in the function<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Jump Discontinuity:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Left-hand and right-hand limits exist but are not equal<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Function \"jumps\" from one value to another<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Infinite Discontinuity:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Function approaches infinity as it nears the point<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Often associated with vertical asymptotes<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identifying Discontinuities:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Evaluate function at the point<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate left-hand and right-hand limits<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Compare limits to function value<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170573361594\">For [latex]f(x)=\\begin{cases} x^2 &amp; \\text{ if } \\, x \\ne 1 \\\\ 3 &amp; \\text{ if } \\, x = 1 \\end{cases}[\/latex], decide whether [latex]f[\/latex] is continuous at [latex]1[\/latex].<\/p>\r\nIf [latex]f[\/latex] is not continuous at [latex]1[\/latex], classify the discontinuity as removable, jump, or infinite.\r\n\r\n[reveal-answer q=\"3388654\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"3388654\"]\r\n<p id=\"fs-id1170573436674\">If the function is discontinuous at [latex]1[\/latex], look at [latex]\\underset{x\\to 1}{\\lim}f(x)[\/latex] and use the definition to determine the type of discontinuity.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170573355512\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573355512\"]\r\n<p id=\"fs-id1170573355512\">Discontinuous at [latex]1[\/latex]; removable<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1170571120474\">State the interval(s) over which the function [latex]f(x)=\\sqrt{x+3}[\/latex] is continuous.<\/p>\r\n[reveal-answer q=\"fs-id1170571287079\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571287079\"]\r\n<p id=\"fs-id1170571287079\">[latex][-3,+\\infty)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Continuity Over an Interval<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Continuity on Open Intervals:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Function is continuous at every point within [latex](a, b)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Continuity on Closed Intervals:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Function is continuous on [latex](a, b)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Right-continuous at [latex]a[\/latex]: [latex]\\lim_{x \\to a^+} f(x) = f(a)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Left-continuous at [latex]b[\/latex]: [latex]\\lim_{x \\to b^-} f(x) = f(b)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Half-Open Intervals:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For [latex](a, b][\/latex]: Continuous on [latex](a, b) [\/latex] and left-continuous at [latex]b[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For [a, b): Continuous on [latex](a, b)[\/latex] and right-continuous at [latex]a[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determining Continuity:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Check domain of the function<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Analyze behavior at endpoint(s) for closed intervals<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Consider discontinuities within the interval<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Determine the interval(s) of continuity for the function:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]f(x) = \\begin{cases} \\sqrt{x+1} &amp; \\text{if } x &lt; 3 \\\\ \\frac{x-3}{x-2} &amp; \\text{if } x \\geq 3 \\end{cases}[\/latex]<\/p>\r\n\r\n[reveal-answer q=\"627478\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"627478\"]\r\n\r\nAnalyze the first piece: [latex]\\sqrt{x+1}[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: [latex]x \\geq -1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Continuous on [latex][-1, 3)[\/latex]<\/li>\r\n<\/ul>\r\nAnalyze the second piece: [latex]\\frac{x-3}{x-2}[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain: [latex]x \\neq 2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Continuous on [latex][3, 2)[\/latex] and [latex](2, \u221e)[\/latex]<\/li>\r\n<\/ul>\r\nCheck continuity at [latex]x = 3[\/latex] (transition point):\r\n<ul>\r\n \t<li>Left limit: [latex]\\lim_{x \\to 3^-} \\sqrt{x+1} = 2[\/latex]<\/li>\r\n \t<li>Right limit: [latex]\\lim_{x \\to 3^+} \\frac{x-3}{x-2} = 0[\/latex]<\/li>\r\n \t<li>[latex]f(3) = 0[\/latex]<\/li>\r\n \t<li>The function is continuous at [latex]x = 3[\/latex].<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, [latex]f(x) [\/latex]is continuous on the intervals [latex][-1, 2)[\/latex] and [latex](2, \u221e)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Composite Function Theorem and The Intermediate Value Theorem<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Composite Function Theorem:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">If [latex]f(x)[\/latex] is continuous at [latex]L[\/latex] and [latex]\\lim_{x \\to a} g(x) = L[\/latex], then: [latex]\\lim_{x \\to a} f(g(x)) = f(\\lim_{x \\to a} g(x)) = f(L)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Helps expand our ability to compute limits<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Demonstrates continuity of trigonometric functions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Continuity of Trigonometric Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">All trigonometric functions are continuous over their entire domains<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Proof uses Composite Function Theorem and continuity of [latex]\\sin x[\/latex] and [latex]\\cos x[\/latex] at [latex]0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Intermediate Value Theorem (IVT):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Applies to functions continuous over closed, bounded intervals [latex][a,b][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]z[\/latex] is between [latex]f(a)[\/latex] and [latex]f(b)[\/latex], there exists [latex]c[\/latex] in [latex][a,b][\/latex] where [latex]f(c) = z[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Useful for proving existence of solutions (e.g., zeros of functions)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Applications of IVT:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Proving existence of zeros for continuous functions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Cannot be used to prove non-existence of zeros or other values<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170573586819\">For [latex]f(x)=1\/x, \\, f(-1)=-1&lt;0[\/latex] and [latex]f(1)=1&gt;0[\/latex]. Can we conclude that [latex]f(x)[\/latex] has a zero in the interval [latex][-1,1][\/latex]?<\/p>\r\n[reveal-answer q=\"fs-id1170571138880\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571138880\"]\r\n<p id=\"fs-id1170571138880\">No. The function is not continuous over [latex][-1,1][\/latex]. The Intermediate Value Theorem does not apply here.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Determine whether a function is continuous at a number.<\/li>\n<li>Determine the input values for which a function is discontinuous.<\/li>\n<\/ul>\n<\/section>\n<h2>Continuity<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">A function [latex]f(x)[\/latex] is continuous at [latex]x = a[\/latex] if:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">f(a) is defined<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} f(x)[\/latex] exists<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} f(x) = f(a)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Types of Discontinuities:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Removable (hole in the graph)<\/li>\n<li class=\"whitespace-normal break-words\">Jump (function value jumps at a point)<\/li>\n<li class=\"whitespace-normal break-words\">Infinite (function approaches infinity near a point)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Continuity of Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Polynomials are continuous everywhere<\/li>\n<li class=\"whitespace-normal break-words\">Rational functions are continuous except where denominator is zero<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Analyzing Continuity:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Check if function is defined at the point<\/li>\n<li class=\"whitespace-normal break-words\">Evaluate the limit as [latex]x[\/latex] approaches the point<\/li>\n<li class=\"whitespace-normal break-words\">Compare the limit value with the function value at the point<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170573406319\">Using the definition, determine whether the function [latex]f(x)=\\begin{cases} 2x+1 & \\text{ if } \\, x < 1 \\\\ 2 & \\text{ if } \\, x = 1 \\\\ -x+4 & \\text{ if } \\, x > 1 \\end{cases}[\/latex] is continuous at [latex]x=1[\/latex].<\/p>\n<p>If the function is not continuous at [latex]1[\/latex], indicate the condition for continuity at a point that fails to hold.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q98945562\">Hint<\/button><\/p>\n<div id=\"q98945562\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170573418814\">Check each condition of the definition.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170573402451\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170573402451\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170573402451\">[latex]f[\/latex] is not continuous at [latex]1[\/latex] because [latex]f(1)=2\\ne 3=\\underset{x\\to 1}{\\lim}f(x)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170570999873\">For what values of [latex]x[\/latex] is [latex]f(x)=3x^4-4x^2[\/latex] continuous?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170573246209\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170573246209\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170573246209\">[latex]f(x)[\/latex] is continuous at every real number.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Types of Discontinuities<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Three Main Types of Discontinuities:\n<ul>\n<li class=\"whitespace-normal break-words\">Removable (hole in the graph)<\/li>\n<li class=\"whitespace-normal break-words\">Jump (function value jumps at a point)<\/li>\n<li class=\"whitespace-normal break-words\">Infinite (function approaches infinity near a point)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Removable Discontinuity:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Limit exists at the point, but function value is different or undefined<\/li>\n<li class=\"whitespace-normal break-words\">Graphically appears as a hole in the function<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Jump Discontinuity:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Left-hand and right-hand limits exist but are not equal<\/li>\n<li class=\"whitespace-normal break-words\">Function &#8220;jumps&#8221; from one value to another<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Infinite Discontinuity:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Function approaches infinity as it nears the point<\/li>\n<li class=\"whitespace-normal break-words\">Often associated with vertical asymptotes<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Identifying Discontinuities:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Evaluate function at the point<\/li>\n<li class=\"whitespace-normal break-words\">Calculate left-hand and right-hand limits<\/li>\n<li class=\"whitespace-normal break-words\">Compare limits to function value<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170573361594\">For [latex]f(x)=\\begin{cases} x^2 & \\text{ if } \\, x \\ne 1 \\\\ 3 & \\text{ if } \\, x = 1 \\end{cases}[\/latex], decide whether [latex]f[\/latex] is continuous at [latex]1[\/latex].<\/p>\n<p>If [latex]f[\/latex] is not continuous at [latex]1[\/latex], classify the discontinuity as removable, jump, or infinite.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q3388654\">Hint<\/button><\/p>\n<div id=\"q3388654\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170573436674\">If the function is discontinuous at [latex]1[\/latex], look at [latex]\\underset{x\\to 1}{\\lim}f(x)[\/latex] and use the definition to determine the type of discontinuity.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170573355512\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170573355512\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170573355512\">Discontinuous at [latex]1[\/latex]; removable<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571120474\">State the interval(s) over which the function [latex]f(x)=\\sqrt{x+3}[\/latex] is continuous.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571287079\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571287079\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571287079\">[latex][-3,+\\infty)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Continuity Over an Interval<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Continuity on Open Intervals:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Function is continuous at every point within [latex](a, b)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Continuity on Closed Intervals:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Function is continuous on [latex](a, b)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Right-continuous at [latex]a[\/latex]: [latex]\\lim_{x \\to a^+} f(x) = f(a)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Left-continuous at [latex]b[\/latex]: [latex]\\lim_{x \\to b^-} f(x) = f(b)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Half-Open Intervals:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex](a, b][\/latex]: Continuous on [latex](a, b)[\/latex] and left-continuous at [latex]b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">For [a, b): Continuous on [latex](a, b)[\/latex] and right-continuous at [latex]a[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Determining Continuity:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Check domain of the function<\/li>\n<li class=\"whitespace-normal break-words\">Analyze behavior at endpoint(s) for closed intervals<\/li>\n<li class=\"whitespace-normal break-words\">Consider discontinuities within the interval<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Determine the interval(s) of continuity for the function:<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]f(x) = \\begin{cases} \\sqrt{x+1} & \\text{if } x < 3 \\\\ \\frac{x-3}{x-2} & \\text{if } x \\geq 3 \\end{cases}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q627478\">Show Answer<\/button><\/p>\n<div id=\"q627478\" class=\"hidden-answer\" style=\"display: none\">\n<p>Analyze the first piece: [latex]\\sqrt{x+1}[\/latex]<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: [latex]x \\geq -1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Continuous on [latex][-1, 3)[\/latex]<\/li>\n<\/ul>\n<p>Analyze the second piece: [latex]\\frac{x-3}{x-2}[\/latex]<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain: [latex]x \\neq 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Continuous on [latex][3, 2)[\/latex] and [latex](2, \u221e)[\/latex]<\/li>\n<\/ul>\n<p>Check continuity at [latex]x = 3[\/latex] (transition point):<\/p>\n<ul>\n<li>Left limit: [latex]\\lim_{x \\to 3^-} \\sqrt{x+1} = 2[\/latex]<\/li>\n<li>Right limit: [latex]\\lim_{x \\to 3^+} \\frac{x-3}{x-2} = 0[\/latex]<\/li>\n<li>[latex]f(3) = 0[\/latex]<\/li>\n<li>The function is continuous at [latex]x = 3[\/latex].<\/li>\n<\/ul>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, [latex]f(x)[\/latex]is continuous on the intervals [latex][-1, 2)[\/latex] and [latex](2, \u221e)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Composite Function Theorem and The Intermediate Value Theorem<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Composite Function Theorem:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]f(x)[\/latex] is continuous at [latex]L[\/latex] and [latex]\\lim_{x \\to a} g(x) = L[\/latex], then: [latex]\\lim_{x \\to a} f(g(x)) = f(\\lim_{x \\to a} g(x)) = f(L)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Helps expand our ability to compute limits<\/li>\n<li class=\"whitespace-normal break-words\">Demonstrates continuity of trigonometric functions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Continuity of Trigonometric Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">All trigonometric functions are continuous over their entire domains<\/li>\n<li class=\"whitespace-normal break-words\">Proof uses Composite Function Theorem and continuity of [latex]\\sin x[\/latex] and [latex]\\cos x[\/latex] at [latex]0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Intermediate Value Theorem (IVT):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Applies to functions continuous over closed, bounded intervals [latex][a,b][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]z[\/latex] is between [latex]f(a)[\/latex] and [latex]f(b)[\/latex], there exists [latex]c[\/latex] in [latex][a,b][\/latex] where [latex]f(c) = z[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Useful for proving existence of solutions (e.g., zeros of functions)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications of IVT:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Proving existence of zeros for continuous functions<\/li>\n<li class=\"whitespace-normal break-words\">Cannot be used to prove non-existence of zeros or other values<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170573586819\">For [latex]f(x)=1\/x, \\, f(-1)=-1<0[\/latex] and [latex]f(1)=1>0[\/latex]. Can we conclude that [latex]f(x)[\/latex] has a zero in the interval [latex][-1,1][\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571138880\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571138880\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571138880\">No. The function is not continuous over [latex][-1,1][\/latex]. The Intermediate Value Theorem does not apply here.<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":23,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":263,"module-header":"fresh_take","content_attributions":[],"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\/1613"}],"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\/67"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1613\/revisions"}],"predecessor-version":[{"id":5645,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1613\/revisions\/5645"}],"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\/1613\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1613"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1613"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1613"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1613"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}