{"id":1065,"date":"2025-10-15T13:55:25","date_gmt":"2025-10-15T13:55:25","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&p=1065"},"modified":"2025-10-23T13:54:34","modified_gmt":"2025-10-23T13:54:34","slug":"limits-and-continuity-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/limits-and-continuity-learn-it-3\/","title":{"raw":"Limits and Continuity: Learn It 3","rendered":"Limits and Continuity: Learn It 3"},"content":{"raw":"

Continuity<\/h2>\r\n

In single-variable calculus, we studied continuity of functions and saw how it relied on limits. The definition of continuity for functions of two variables follows the same pattern, with three key conditions that must be satisfied.<\/p>\r\n\r\n

\r\n

Continuity of a Function of One Variable<\/strong><\/p>\r\n

A function [latex]f(x)[\/latex] is continuous at point [latex]x=a[\/latex] if three conditions hold:<\/p>\r\n\r\n

    \r\n \t
  1. [latex]f(a)[\/latex] exists.<\/li>\r\n \t
  2. [latex]\\underset{x\\to a}{\\text{lim}}f(x)[\/latex] exists.<\/li>\r\n \t
  3. [latex]\\underset{x\\to a}{\\text{lim}}f(x)=f(a)[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>\r\n

    These three conditions are necessary for continuity of a function of two variables as well.<\/p>\r\n\r\n

    \r\n

    continuity of a function of two variables<\/h3>\r\nA function [latex]f\\,(x,\\ y)[\/latex] is continuous<\/strong> at a point [latex](a,\\ b)[\/latex] in its domain if:\r\n
      \r\n \t
    1. [latex]f\\,(a,\\ b)[\/latex] exists.<\/li>\r\n \t
    2. [latex]\\displaystyle{\\lim_{(x,\\ y)\\to{(a,\\ b)}}}f\\,(x,\\ y)[\/latex] exists.<\/li>\r\n \t
    3. [latex]\\displaystyle{\\lim_{(x,\\ y)\\to(a,\\ b)}}f\\,(x,\\ y)=f\\,(a,\\ b)[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
      Show that the function [latex]f\\,(x,\\ y)=\\frac{3x+2y}{x+y+1}[\/latex] is continuous at point [latex](5,-3)[\/latex].[reveal-answer q=\"fs-id1667793753114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1667793753114\"]\r\n
      There are three conditions to be satisfied, per the definition of continuity. In this example, [latex]a=5[\/latex] and [latex]b=-3[\/latex].<\/div>\r\n
        \r\n \t
      1. [latex]f\\,(a,\\ b)[\/latex]\u00a0exists. This is true because the domain of the function [latex]f[\/latex]\u00a0consists of those ordered pairs for which the denominator is nonzero (i.e., [latex]x+y+1\\neq{0}[\/latex]). Point [latex](5,-3)[\/latex]\u00a0satisfies this condition. Furthermore,\r\n
        [latex]f\\,(a,\\ b)=f\\,(5,-3)=\\frac{3(5)+2(-3)}{5+(-3)+1}=\\frac{15-6}{2+1}=3.[\/latex]<\/div>\r\n <\/li>\r\n \t
      2. [latex]\\displaystyle\\lim_{(x,\\ y)\\to(a,\\ b)}f\\,(x,\\ y)[\/latex] exists. This is also true:\r\n
        [latex]\\begin{array}{ccc}\\hfill {\\displaystyle\\lim_{(x,\\ y)\\to(a,\\ b)}f\\,(x,\\ y)} & =\\hfill & {\\displaystyle\\lim_{(x,\\ y)\\to(5,\\ -3)}\\frac{3x+2y}{x+y+1}}\\hfill \\\\ \\hfill & =\\hfill & {\\frac{\\displaystyle\\lim_{(x,\\ y)\\to(5,\\ -3)}(3x+2y)}{\\displaystyle\\lim_{(x,\\ y)\\to(5,\\ -3)}(x+y+1)}}\\hfill \\\\ \\hfill & =\\hfill & {\\frac{15-6}{5-3+1}}\\hfill \\\\ \\hfill & =\\hfill & {3.}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div><\/li>\r\n \t
      3. [latex]\\displaystyle\\lim_{(x,\\ y)\\to(a,\\ b)}f\\,(x,\\ y)=f\\,(a,\\ b)[\/latex].\u00a0This is true because we have just shown that both sides of this equation equal three.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n

        Properties of Continuous Functions<\/h3>\r\n

        Continuity can also be defined using the epsilon-delta definition. A function of two variables is continuous at a point [latex](x_0,y_0)[\/latex] in its domain if for every [latex]\\epsilon>0[\/latex] there exists [latex]\\delta>0[\/latex] such that, whenever [latex]\\sqrt{(x-x_0)^2+(y-y_0)^2}<\\delta[\/latex], we have [latex]|f(x,y)-f(x_0,y_0)|<\\epsilon[\/latex].<\/p>\r\n

        This definition can be combined with the epsilon-delta definition of continuity for functions of one variable to prove the following theorems.<\/p>\r\n\r\n

        \r\n

        sum of continuous functions<\/h3>\r\nIf [latex]f(x,y)[\/latex] is continuous at\u00a0[latex](x_0,y_0)[\/latex], and\u00a0[latex]g(x,y)[\/latex] is continuous at\u00a0[latex](x_0,y_0)[\/latex], then\u00a0[latex]f(x,y)+g(x,y)[\/latex] is continuous at\u00a0[latex](x_0,y_0)[\/latex].\r\n\r\n<\/section>
        \r\n

        product of continuous functions<\/h3>\r\nIf [latex]g(x)[\/latex] is continuous at\u00a0[latex]x_0[\/latex], and\u00a0[latex]h(y)[\/latex] is continuous at\u00a0[latex]y_0[\/latex], then\u00a0[latex]f(x,y)=g(x)h(y)[\/latex] is continuous at\u00a0[latex](x_0,y_0)[\/latex].\r\n\r\n<\/section>
        \r\n

        composition of continuous functions<\/h3>\r\nLet [latex]g[\/latex] be a function of two variables continuous at [latex](x_0,y_0)\\in{D}[\/latex], and define [latex]z_0=g(x_0,y_0)[\/latex]. Let [latex]f[\/latex] be a function of one variable continuous at [latex]z_0[\/latex]. Then [latex]f\\circ g[\/latex] is continuous at [latex](x_0,y_0)[\/latex].\r\n\r\n<\/section>\r\n\r\n[caption id=\"attachment_979\" align=\"aligncenter\" width=\"856\"]\"A Figure 1.\u00a0The composition of two continuous functions is continuous.[\/caption]\r\n

        We can use these theorems to show continuity of functions in the following example.<\/p>\r\n\r\n

        Show that the functions [latex]f\\,(x,\\ y)=4x^{3}y^{2}[\/latex] and [latex]g\\,(x,\\ y)=\\cos{(4x^{3}y^{2})}[\/latex] are continuous everywhere.[reveal-answer q=\"fs-id2667993733114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id2667993733114\"]\r\n\r\nThe polynomials [latex]g\\,(x)=4x^{3}[\/latex] and [latex]h\\,(y)=y^{2}[\/latex]\u00a0are continuous at every real number, and therefore by the product of continuous functions theorem, [latex]f\\,(x,\\ y)=4x^{3}y^{2}[\/latex] is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane. Since [latex]f\\,(x,\\ y)=4x^{3}y^{2}[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane and [latex]g\\,(x)=\\cos{x}[\/latex]\u00a0is continuous at every real number [latex]x[\/latex],\u00a0the continuity of the composition of functions tells us that [latex]g\\,(x,\\ y)=\\cos{(4x^{3}y^{2})}[\/latex] is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"

        Continuity<\/h2>\n

        In single-variable calculus, we studied continuity of functions and saw how it relied on limits. The definition of continuity for functions of two variables follows the same pattern, with three key conditions that must be satisfied.<\/p>\n

        \n

        Continuity of a Function of One Variable<\/strong><\/p>\n

        A function [latex]f(x)[\/latex] is continuous at point [latex]x=a[\/latex] if three conditions hold:<\/p>\n

          \n
        1. [latex]f(a)[\/latex] exists.<\/li>\n
        2. [latex]\\underset{x\\to a}{\\text{lim}}f(x)[\/latex] exists.<\/li>\n
        3. [latex]\\underset{x\\to a}{\\text{lim}}f(x)=f(a)[\/latex].<\/li>\n<\/ol>\n<\/section>\n

          These three conditions are necessary for continuity of a function of two variables as well.<\/p>\n

          \n

          continuity of a function of two variables<\/h3>\n

          A function [latex]f\\,(x,\\ y)[\/latex] is continuous<\/strong> at a point [latex](a,\\ b)[\/latex] in its domain if:<\/p>\n

            \n
          1. [latex]f\\,(a,\\ b)[\/latex] exists.<\/li>\n
          2. [latex]\\displaystyle{\\lim_{(x,\\ y)\\to{(a,\\ b)}}}f\\,(x,\\ y)[\/latex] exists.<\/li>\n
          3. [latex]\\displaystyle{\\lim_{(x,\\ y)\\to(a,\\ b)}}f\\,(x,\\ y)=f\\,(a,\\ b)[\/latex].<\/li>\n<\/ol>\n<\/section>\n
            Show that the function [latex]f\\,(x,\\ y)=\\frac{3x+2y}{x+y+1}[\/latex] is continuous at point [latex](5,-3)[\/latex].<\/p>\n