{"id":2399,"date":"2024-05-22T13:55:17","date_gmt":"2024-05-22T13:55:17","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=2399"},"modified":"2025-08-17T22:29:13","modified_gmt":"2025-08-17T22:29:13","slug":"continuity-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/continuity-learn-it-2\/","title":{"raw":"Continuity: Learn It 2","rendered":"Continuity: Learn It 2"},"content":{"raw":"<h2>Types of Discontinuities<\/h2>\r\n<p id=\"fs-id1170573435662\">As we have seen in the earlier examples, discontinuities take on several different appearances. We classify the types of discontinuities we have seen thus far as removable discontinuities, infinite discontinuities, or jump discontinuities.<\/p>\r\n<ul>\r\n\t<li>A <strong>removable discontinuity<\/strong> is a discontinuity for which there is a hole in the graph<\/li>\r\n\t<li>A <strong>jump discontinuity<\/strong> is a noninfinite discontinuity for which the sections of the function do not meet up<\/li>\r\n\t<li>An <strong>infinite discontinuity<\/strong> is a discontinuity located at a vertical asymptote.<\/li>\r\n<\/ul>\r\n<p>The figure below illustrates the differences in these types of discontinuities.<\/p>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203514\/CNX_Calc_Figure_02_04_006.jpg\" alt=\"&quot;&quot;Three\" width=\"975\" height=\"315\" \/> Diagram showing different types of discontinuity[\/caption]\r\n\r\n<p id=\"fs-id1170573400557\">These three discontinuities are formally defined as follows:<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>types of discontinuities<\/h3>\r\n<p>If [latex]f(x)[\/latex] is discontinuous at [latex]a[\/latex], then<\/p>\r\n<ol id=\"fs-id1170573216342\">\r\n\t<li>[latex]f[\/latex] has a <strong>removable discontinuity<\/strong> at [latex]a[\/latex] if [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] exists. <br \/>\r\n(Note: When we state that [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] exists, we mean that [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex], where [latex]L[\/latex] is a real number.)<\/li>\r\n\t<li>[latex]f[\/latex] has a <strong>jump discontinuity<\/strong> at [latex]a[\/latex] if [latex]\\underset{x\\to a^-}{\\lim}f(x)[\/latex] and [latex]\\underset{x\\to a^+}{\\lim}f(x)[\/latex] both exist, but [latex]\\underset{x\\to a^-}{\\lim}f(x)\\ne \\underset{x\\to a^+}{\\lim}f(x)[\/latex]. <br \/>\r\n(Note: When we state that [latex]\\underset{x\\to a^-}{\\lim}f(x)[\/latex] and [latex]\\underset{x\\to a^+}{\\lim}f(x)[\/latex] both exist, we mean that both are real-valued and that neither take on the values [latex]\\pm \\infty[\/latex].)<\/li>\r\n\t<li>[latex]f[\/latex] has an <strong>infinite discontinuity<\/strong> at [latex]a[\/latex] if [latex]\\underset{x\\to a^-}{\\lim}f(x)=\\pm \\infty [\/latex] or [latex]\\underset{x\\to a^+}{\\lim}f(x)=\\pm \\infty[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p>Although these terms provide a handy way of describing three common types of discontinuities, keep in mind that not all discontinuities fit neatly into these categories.<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170573424344\">In an earlier example, we showed that [latex]f(x)=\\dfrac{x^2-4}{x-2}[\/latex] is discontinuous at [latex]x=2[\/latex]. Classify this discontinuity as removable, jump, or infinite.<\/p>\r\n<p>[reveal-answer q=\"fs-id1170573402024\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170573402024\"]<\/p>\r\n<p id=\"fs-id1170573402024\">To classify the discontinuity at [latex]2[\/latex] we must evaluate [latex]\\underset{x\\to 2}{\\lim}f(x)[\/latex]:<\/p>\r\n<div id=\"fs-id1170571103554\" class=\"equation unnumbered\" style=\"text-align: left;\">[latex]\\begin{array}{cc}\\underset{x\\to 2}{\\lim}f(x) &amp; =\\underset{x\\to 2}{\\lim}\\frac{x^2-4}{x-2} \\\\ &amp; =\\underset{x\\to 2}{\\lim}\\frac{(x-2)(x+2)}{x-2} \\\\ &amp; =\\underset{x\\to 2}{\\lim}(x+2) \\\\ &amp; = 4 \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170571136107\">Since [latex]f[\/latex] is discontinuous at [latex]2[\/latex] and [latex]\\underset{x\\to 2}{\\lim}f(x)[\/latex] exists, [latex]f[\/latex] has a removable discontinuity at [latex]x=2[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170573586256\">In an earlier example, we showed that [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 discontinuous at [latex]x=3[\/latex]. Classify this discontinuity as removable, jump, or infinite.<\/p>\r\n<p>[reveal-answer q=\"fs-id1170571095481\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170571095481\"]<\/p>\r\n<p id=\"fs-id1170571095481\">Earlier, we showed that [latex]f[\/latex] is discontinuous at [latex]3[\/latex] because [latex]\\underset{x\\to 3}{\\lim}f(x)[\/latex] does not exist. However, since [latex]\\underset{x\\to 3^-}{\\lim}f(x)=-5[\/latex] and [latex]\\underset{x\\to 3^+}{\\lim}f(x)=4[\/latex] both exist, we conclude that the function has a jump discontinuity at [latex]3[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170573732461\">Determine whether [latex]f(x)=\\dfrac{x+2}{x+1}[\/latex] is continuous at [latex]\u22121[\/latex]. If the function is discontinuous at [latex]\u22121[\/latex], classify the discontinuity as removable, jump, or infinite.<\/p>\r\n<p>[reveal-answer q=\"fs-id1170571000111\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170571000111\"]<\/p>\r\n<p id=\"fs-id1170571000111\">The function value [latex]f(-1)[\/latex] is undefined. Therefore, the function is not continuous at [latex]\u22121[\/latex]. To determine the type of discontinuity, we must determine the limit at [latex]\u22121[\/latex]. We see that [latex]\\underset{x\\to -1^-}{\\lim}\\frac{x+2}{x+1}=\u2212\\infty [\/latex] and [latex]\\underset{x\\to -1^+}{\\lim}\\frac{x+2}{x+1}=+\\infty [\/latex]. Therefore, the function has an infinite discontinuity at [latex]\u22121[\/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=529&amp;end=591&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"266834\"]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\/2.4Continuity529to591_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>","rendered":"<h2>Types of Discontinuities<\/h2>\n<p id=\"fs-id1170573435662\">As we have seen in the earlier examples, discontinuities take on several different appearances. We classify the types of discontinuities we have seen thus far as removable discontinuities, infinite discontinuities, or jump discontinuities.<\/p>\n<ul>\n<li>A <strong>removable discontinuity<\/strong> is a discontinuity for which there is a hole in the graph<\/li>\n<li>A <strong>jump discontinuity<\/strong> is a noninfinite discontinuity for which the sections of the function do not meet up<\/li>\n<li>An <strong>infinite discontinuity<\/strong> is a discontinuity located at a vertical asymptote.<\/li>\n<\/ul>\n<p>The figure below illustrates the differences in these types of discontinuities.<\/p>\n<figure style=\"width: 975px\" 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\/11203514\/CNX_Calc_Figure_02_04_006.jpg\" alt=\"&quot;&quot;Three\" width=\"975\" height=\"315\" \/><figcaption class=\"wp-caption-text\">Diagram showing different types of discontinuity<\/figcaption><\/figure>\n<p id=\"fs-id1170573400557\">These three discontinuities are formally defined as follows:<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>types of discontinuities<\/h3>\n<p>If [latex]f(x)[\/latex] is discontinuous at [latex]a[\/latex], then<\/p>\n<ol id=\"fs-id1170573216342\">\n<li>[latex]f[\/latex] has a <strong>removable discontinuity<\/strong> at [latex]a[\/latex] if [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] exists. <br \/>\n(Note: When we state that [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] exists, we mean that [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex], where [latex]L[\/latex] is a real number.)<\/li>\n<li>[latex]f[\/latex] has a <strong>jump discontinuity<\/strong> at [latex]a[\/latex] if [latex]\\underset{x\\to a^-}{\\lim}f(x)[\/latex] and [latex]\\underset{x\\to a^+}{\\lim}f(x)[\/latex] both exist, but [latex]\\underset{x\\to a^-}{\\lim}f(x)\\ne \\underset{x\\to a^+}{\\lim}f(x)[\/latex]. <br \/>\n(Note: When we state that [latex]\\underset{x\\to a^-}{\\lim}f(x)[\/latex] and [latex]\\underset{x\\to a^+}{\\lim}f(x)[\/latex] both exist, we mean that both are real-valued and that neither take on the values [latex]\\pm \\infty[\/latex].)<\/li>\n<li>[latex]f[\/latex] has an <strong>infinite discontinuity<\/strong> at [latex]a[\/latex] if [latex]\\underset{x\\to a^-}{\\lim}f(x)=\\pm \\infty[\/latex] or [latex]\\underset{x\\to a^+}{\\lim}f(x)=\\pm \\infty[\/latex].<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox proTip\">\n<p>Although these terms provide a handy way of describing three common types of discontinuities, keep in mind that not all discontinuities fit neatly into these categories.<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170573424344\">In an earlier example, we showed that [latex]f(x)=\\dfrac{x^2-4}{x-2}[\/latex] is discontinuous at [latex]x=2[\/latex]. Classify this discontinuity as removable, jump, or infinite.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170573402024\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170573402024\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170573402024\">To classify the discontinuity at [latex]2[\/latex] we must evaluate [latex]\\underset{x\\to 2}{\\lim}f(x)[\/latex]:<\/p>\n<div id=\"fs-id1170571103554\" class=\"equation unnumbered\" style=\"text-align: left;\">[latex]\\begin{array}{cc}\\underset{x\\to 2}{\\lim}f(x) & =\\underset{x\\to 2}{\\lim}\\frac{x^2-4}{x-2} \\\\ & =\\underset{x\\to 2}{\\lim}\\frac{(x-2)(x+2)}{x-2} \\\\ & =\\underset{x\\to 2}{\\lim}(x+2) \\\\ & = 4 \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170571136107\">Since [latex]f[\/latex] is discontinuous at [latex]2[\/latex] and [latex]\\underset{x\\to 2}{\\lim}f(x)[\/latex] exists, [latex]f[\/latex] has a removable discontinuity at [latex]x=2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170573586256\">In an earlier example, we showed that [latex]f(x)=\\begin{cases} -x^2+4 & \\text{ if } \\, x \\le 3 \\\\ 4x-8 & \\text{ if } \\, x > 3 \\end{cases}[\/latex] is discontinuous at [latex]x=3[\/latex]. Classify this discontinuity as removable, jump, or infinite.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571095481\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571095481\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571095481\">Earlier, we showed that [latex]f[\/latex] is discontinuous at [latex]3[\/latex] because [latex]\\underset{x\\to 3}{\\lim}f(x)[\/latex] does not exist. However, since [latex]\\underset{x\\to 3^-}{\\lim}f(x)=-5[\/latex] and [latex]\\underset{x\\to 3^+}{\\lim}f(x)=4[\/latex] both exist, we conclude that the function has a jump discontinuity at [latex]3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170573732461\">Determine whether [latex]f(x)=\\dfrac{x+2}{x+1}[\/latex] is continuous at [latex]\u22121[\/latex]. If the function is discontinuous at [latex]\u22121[\/latex], classify the discontinuity as removable, jump, or infinite.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571000111\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571000111\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571000111\">The function value [latex]f(-1)[\/latex] is undefined. Therefore, the function is not continuous at [latex]\u22121[\/latex]. To determine the type of discontinuity, we must determine the limit at [latex]\u22121[\/latex]. We see that [latex]\\underset{x\\to -1^-}{\\lim}\\frac{x+2}{x+1}=\u2212\\infty[\/latex] and [latex]\\underset{x\\to -1^+}{\\lim}\\frac{x+2}{x+1}=+\\infty[\/latex]. Therefore, the function has an infinite discontinuity at [latex]\u22121[\/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=529&amp;end=591&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=\"q266834\">Closed Captioning and Transcript Information for Video<\/button><\/p>\n<div id=\"q266834\" 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\/2.4Continuity529to591_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","protected":false},"author":15,"menu_order":11,"template":"","meta":{"_candela_citation":"[]","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":[],"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\/2399"}],"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\/15"}],"version-history":[{"count":12,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2399\/revisions"}],"predecessor-version":[{"id":4755,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2399\/revisions\/4755"}],"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\/2399\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=2399"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=2399"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=2399"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=2399"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}