{"id":3002,"date":"2025-08-15T19:34:32","date_gmt":"2025-08-15T19:34:32","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=3002"},"modified":"2025-08-15T19:34:32","modified_gmt":"2025-08-15T19:34:32","slug":"continuity-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/continuity-learn-it-3\/","title":{"raw":"Continuity: Learn It 3","rendered":"Continuity: Learn It 3"},"content":{"raw":"

Recognizing Continuous and Discontinuous Real-Number Functions<\/h2>\r\nMany of the functions we have encountered in earlier chapters are continuous everywhere. They never have a hole in them, and they never jump from one value to the next. For all of these functions, the limit of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] is the same as the value of [latex]f\\left(x\\right)[\/latex] when [latex]x=a[\/latex]. So [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=f\\left(a\\right)[\/latex]. There are some functions that are continuous everywhere and some that are only continuous where they are defined on their domain because they are not defined for all real numbers.\r\n\r\n
\r\n

continuous functions<\/h3>\r\nThe following functions are continuous everywhere:\r\n<\/colgroup>\r\n\r\n\r\n\r\n\r\n\r\n
Polynomial functions<\/td>\r\nEx: [latex]f\\left(x\\right)={x}^{4}-9{x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n
Exponential functions<\/td>\r\nEx: [latex]f\\left(x\\right)={4}^{x+2}-5[\/latex]<\/td>\r\n<\/tr>\r\n
Sine functions<\/td>\r\nEx: [latex]f\\left(x\\right)=\\sin \\left(2x\\right)-4[\/latex]<\/td>\r\n<\/tr>\r\n
Cosine functions<\/td>\r\nEx: [latex]f\\left(x\\right)=-\\cos \\left(x+\\frac{\\pi }{3}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe following functions are continuous everywhere they are defined on their domain:\r\n<\/colgroup>\r\n\r\n\r\n\r\n\r\n
Logarithmic functions<\/td>\r\nEx: [latex]f\\left(x\\right)=2\\mathrm{ln}\\left(x\\right)[\/latex] , [latex]x>0[\/latex]<\/td>\r\n<\/tr>\r\n
Tangent functions<\/td>\r\nEx: [latex]f\\left(x\\right)=\\tan \\left(x\\right)+2[\/latex], [latex]x\\ne \\frac{\\pi }{2}+k\\pi [\/latex], [latex]k[\/latex] is an integer<\/td>\r\n<\/tr>\r\n
Rational functions<\/td>\r\nEx: [latex]f\\left(x\\right)=\\frac{{x}^{2}-25}{x - 7}[\/latex], [latex]x\\ne 7[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section>
How To: Given a function [latex]\\begin{align}f\\left(x\\right)\\end{align}[\/latex], determine if the function is continuous at [latex]\\begin{align}x=a\\end{align}[\/latex].<\/strong>\r\n
    \r\n \t
  1. Check Condition 1: [latex]f\\left(a\\right)[\/latex] exists.<\/li>\r\n \t
  2. Check Condition 2: [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exists at [latex]x=a[\/latex].<\/li>\r\n \t
  3. Check Condition 3: [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=f\\left(a\\right)[\/latex].<\/li>\r\n \t
  4. If all three conditions are satisfied, the function is continuous at [latex]x=a[\/latex]. If any one of the conditions is not satisfied, the function is not continuous at [latex]x=a[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
    Determine whether the function [latex]f\\left(x\\right)=\\begin{cases}4x, \\hfill& x\\leq 3 \\\\ 8+x, \\hfill& x>3\\end{cases}[\/latex] is continuous at\r\n
      \r\n \t
    1. [latex]x=3[\/latex]<\/li>\r\n \t
    2. [latex]x=\\frac{8}{3}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"324975\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"324975\"]\r\n\r\nTo determine if the function [latex]f[\/latex] is continuous at [latex]x=a[\/latex], we will determine if the three conditions of continuity are satisfied at [latex]x=a[\/latex] .\r\n
        \r\n \t
      1. Condition 1: Does [latex]f\\left(a\\right)[\/latex] exist?\r\n
        [latex]\\begin{align}&f\\left(3\\right)=4\\left(3\\right)=12 \\\\ &\\Rightarrow \\text{Condition 1 is satisfied}. \\end{align}[\/latex]<\/div>\r\nCondition 2: Does [latex]\\underset{x\\to 3}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exist?\r\n\r\nTo the left of [latex]x=3[\/latex], [latex]f\\left(x\\right)=4x[\/latex]; to the right of [latex]x=3[\/latex], [latex]f\\left(x\\right)=8+x[\/latex]. We need to evaluate the left- and right-hand limits as [latex]x[\/latex] approaches 1.\r\n
        Left-hand limit: [latex]\\underset{x\\to {3}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {3}^{-}}{\\mathrm{lim}}4\\left(3\\right)=12[\/latex]\r\nRight-hand limit: [latex]\\underset{x\\to {3}^{+}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {3}^{+}}{\\mathrm{lim}}\\left(8+x\\right)=8+3=11[\/latex]<\/div>\r\nBecause [latex]\\underset{x\\to {1}^{-}}{\\mathrm{lim}}f\\left(x\\right)\\ne \\underset{x\\to {1}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex], [latex]\\underset{x\\to 1}{\\mathrm{lim}}f\\left(x\\right)[\/latex] does not exist.\r\n<\/span>\r\n
        [latex]\\Rightarrow \\text{ Condition 2 fails}\\text{.}[\/latex]<\/div>\r\nThere is no need to proceed further. Condition 2 fails at [latex]x=3[\/latex]. If any of the conditions of continuity are not satisfied at [latex]x=3[\/latex], the function [latex]f\\left(x\\right)[\/latex] is not continuous at [latex]x=3[\/latex].<\/li>\r\n \t
      2. [latex]x=\\frac{8}{3}[\/latex]Condition 1: Does [latex]f\\left(\\frac{8}{3}\\right)[\/latex] exist?\r\n
        [latex]\\begin{align}&f\\left(\\frac{8}{3}\\right)=4\\left(\\frac{8}{3}\\right)=\\frac{32}{3} \\\\ &\\Rightarrow \\text{Condition 1 is satisfied}. \\end{align}[\/latex]<\/div>\r\nCondition 2: Does [latex]\\underset{x\\to \\frac{8}{3}}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exist?<\/span>\r\n\r\nTo the left of [latex]x=\\frac{8}{3}[\/latex], [latex]f\\left(x\\right)=4x[\/latex]; to the right of [latex]x=\\frac{8}{3}[\/latex], [latex]f\\left(x\\right)=8+x[\/latex]. We need to evaluate the left- and right-hand limits as [latex]x[\/latex] approaches [latex]\\frac{8}{3}[\/latex].\r\n
        \r\n
        \r\n
        Left-hand limit: [latex]\\underset{x\\to {\\frac{8}{3}}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {\\frac{8}{3}}^{-}}{\\mathrm{lim}}4\\left(\\frac{8}{3}\\right)=\\frac{32}{3}[\/latex]\r\nRight-hand limit: [latex]\\underset{x\\to {\\frac{8}{3}}^{+}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {\\frac{8}{3}}^{+}}{\\mathrm{lim}}\\left(8+x\\right)=8+\\frac{8}{3}=\\frac{32}{3}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\nBecause [latex]\\underset{x\\to \\frac{8}{3}}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exists,\r\n
        [latex]\\Rightarrow \\text{Condition 2 is satisfied}[\/latex].<\/div>\r\nCondition 3: Is [latex]f\\left(\\frac{8}{3}\\right)=\\underset{x\\to \\frac{8}{3}}{\\mathrm{lim}}f\\left(x\\right)?[\/latex]\r\n
        [latex]\\begin{align}f&\\left(\\frac{32}{3}\\right)=\\frac{32}{3}=\\underset{x\\to \\frac{8}{3}}{\\mathrm{lim}}f\\left(x\\right) \\\\ &\\Rightarrow \\text{Condition 3 is satisfied}. \\end{align}[\/latex]<\/div>\r\nBecause all three conditions of continuity are satisfied at [latex]x=\\frac{8}{3}[\/latex], the function [latex]f\\left(x\\right)[\/latex] is continuous at [latex]x=\\frac{8}{3}[\/latex].\r\n
          \r\n \t
        1. [\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/section>
          Determine whether the function [latex]f\\left(x\\right)=\\begin{cases}\\frac{1}{x}, \\hfill& x\\leq 2 \\\\ 9x-17.5, \\hfill& x>2\\end{cases}[\/latex] is continuous at [latex]x=2[\/latex].[reveal-answer q=\"968995\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"968995\"]yes[\/hidden-answer]\r\n\r\n<\/section>
          Determine whether the function [latex]f\\left(x\\right)=\\frac{{x}^{2}-25}{x - 5}[\/latex] is continuous at [latex]x=5[\/latex].\r\n\r\n[reveal-answer q=\"48479\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"48479\"]\r\n\r\nTo determine if the function [latex]f[\/latex] is continuous at [latex]x=5[\/latex], we will determine if the three conditions of continuity are satisfied at [latex]x=5[\/latex].\r\n\r\nCondition 1:\r\n

          [latex]\\begin{gathered}f\\left(5\\right)\\text{ does not exist.} \\\\ \\Rightarrow \\text{Condition 1 fails}.\\end{gathered}[\/latex]<\/p>\r\nThere is no need to proceed further. Condition 2 fails at [latex]x=5[\/latex]. If any of the conditions of continuity are not satisfied at [latex]x=5[\/latex], the function [latex]f[\/latex] is not continuous at [latex]x=5[\/latex].\r\n

          Analysis of the Solution<\/h4>\r\nNotice that for Condition 2 we have\r\n

          [latex]\\begin{align}\\underset{x\\to 5}{\\mathrm{lim}}\\frac{{x}^{2}-25}{x - 5}&=\\underset{x\\to 3}{\\mathrm{lim}}\\frac{\\cancel{\\left(x - 5\\right)}\\left(x+5\\right)}{\\cancel{x - 5}} \\\\ &=\\underset{x\\to 5}{\\mathrm{lim}}\\left(x+5\\right) \\\\ &=5+5=10 \\\\ \\Rightarrow \\text{Conditio}&\\text{n 2 is satisfied}. \\end{align}[\/latex]<\/p>\r\nAt [latex]x=5[\/latex], there exists a removable discontinuity.\r\n\r\n\"Graph\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

          \r\n
          \r\n\r\nDetermine whether the function [latex]f\\left(x\\right)=\\frac{9-{x}^{2}}{{x}^{2}-3x}[\/latex] is continuous at [latex]x=3[\/latex]. If not, state the type of discontinuity.\r\n\r\n[reveal-answer q=\"252333\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"252333\"]\r\n\r\nNo, the function is not continuous at [latex]x=3[\/latex]. There exists a removable discontinuity at [latex]x=3[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>
          [ohm_question hide_question_numbers=1]174101[\/ohm_question]<\/section>","rendered":"

          Recognizing Continuous and Discontinuous Real-Number Functions<\/h2>\n

          Many of the functions we have encountered in earlier chapters are continuous everywhere. They never have a hole in them, and they never jump from one value to the next. For all of these functions, the limit of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] is the same as the value of [latex]f\\left(x\\right)[\/latex] when [latex]x=a[\/latex]. So [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=f\\left(a\\right)[\/latex]. There are some functions that are continuous everywhere and some that are only continuous where they are defined on their domain because they are not defined for all real numbers.<\/p>\n

          \n

          continuous functions<\/h3>\n

          The following functions are continuous everywhere:<\/p>\n\n\n\n<\/colgroup>\n\n\n\n\n\n
          Polynomial functions<\/td>\nEx: [latex]f\\left(x\\right)={x}^{4}-9{x}^{2}[\/latex]<\/td>\n<\/tr>\n
          Exponential functions<\/td>\nEx: [latex]f\\left(x\\right)={4}^{x+2}-5[\/latex]<\/td>\n<\/tr>\n
          Sine functions<\/td>\nEx: [latex]f\\left(x\\right)=\\sin \\left(2x\\right)-4[\/latex]<\/td>\n<\/tr>\n
          Cosine functions<\/td>\nEx: [latex]f\\left(x\\right)=-\\cos \\left(x+\\frac{\\pi }{3}\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

          The following functions are continuous everywhere they are defined on their domain:<\/p>\n\n\n\n<\/colgroup>\n\n\n\n\n
          Logarithmic functions<\/td>\nEx: [latex]f\\left(x\\right)=2\\mathrm{ln}\\left(x\\right)[\/latex] , [latex]x>0[\/latex]<\/td>\n<\/tr>\n
          Tangent functions<\/td>\nEx: [latex]f\\left(x\\right)=\\tan \\left(x\\right)+2[\/latex], [latex]x\\ne \\frac{\\pi }{2}+k\\pi[\/latex], [latex]k[\/latex] is an integer<\/td>\n<\/tr>\n
          Rational functions<\/td>\nEx: [latex]f\\left(x\\right)=\\frac{{x}^{2}-25}{x - 7}[\/latex], [latex]x\\ne 7[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n
          How To: Given a function [latex]\\begin{align}f\\left(x\\right)\\end{align}[\/latex], determine if the function is continuous at [latex]\\begin{align}x=a\\end{align}[\/latex].<\/strong><\/p>\n
            \n
          1. Check Condition 1: [latex]f\\left(a\\right)[\/latex] exists.<\/li>\n
          2. Check Condition 2: [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exists at [latex]x=a[\/latex].<\/li>\n
          3. Check Condition 3: [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=f\\left(a\\right)[\/latex].<\/li>\n
          4. If all three conditions are satisfied, the function is continuous at [latex]x=a[\/latex]. If any one of the conditions is not satisfied, the function is not continuous at [latex]x=a[\/latex].<\/li>\n<\/ol>\n<\/section>\n
            Determine whether the function [latex]f\\left(x\\right)=\\begin{cases}4x, \\hfill& x\\leq 3 \\\\ 8+x, \\hfill& x>3\\end{cases}[\/latex] is continuous at<\/p>\n
              \n
            1. [latex]x=3[\/latex]<\/li>\n
            2. [latex]x=\\frac{8}{3}[\/latex]<\/li>\n<\/ol>\n