Limits and Continuity: Learn It 3

Continuity

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.

Continuity of a Function of One Variable

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

  1. [latex]f(a)[/latex] exists.
  2. [latex]\underset{x\to a}{\text{lim}}f(x)[/latex] exists.
  3. [latex]\underset{x\to a}{\text{lim}}f(x)=f(a)[/latex].

These three conditions are necessary for continuity of a function of two variables as well.

continuity of a function of two variables

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

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

Properties of Continuous Functions

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].

This definition can be combined with the epsilon-delta definition of continuity for functions of one variable to prove the following theorems.

sum of continuous functions

If [latex]f(x,y)[/latex] is continuous at [latex](x_0,y_0)[/latex], and [latex]g(x,y)[/latex] is continuous at [latex](x_0,y_0)[/latex], then [latex]f(x,y)+g(x,y)[/latex] is continuous at [latex](x_0,y_0)[/latex].

product of continuous functions

If [latex]g(x)[/latex] is continuous at [latex]x_0[/latex], and [latex]h(y)[/latex] is continuous at [latex]y_0[/latex], then [latex]f(x,y)=g(x)h(y)[/latex] is continuous at [latex](x_0,y_0)[/latex].

composition of continuous functions

Let [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].

A shape is shown labeled the domain of g with point (x, y) inside of it. From the domain of g there is an arrow marked g pointing to the range of g, which is a straight line with point z on it. The range of g is also marked the domain of f. Then there is another arrow marked f from this shape to a line marked range of f.
Figure 1. The composition of two continuous functions is continuous.

We can use these theorems to show continuity of functions in the following example.

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.