Learning Outcomes
- State the conditions for continuity of a function of two variables.
- Verify the continuity of a function of two variables at a point.
- Calculate the limit of a function of three or more variables and verify the continuity of the function at a point.
In Continuity, we defined the continuity of a function of one variable and saw how it relied on the limit of a function of one variable. In particular, three conditions are necessary for [latex]f\,(x)[/latex] to be continuous at point [latex]x=a[/latex]:
- [latex]f\,(a)[/latex] exists.
- [latex]\displaystyle{\lim_{x\to{a}}}f\,(x)[/latex] exists.
- [latex]\displaystyle{\lim_{x\to{a}}}f\,(x)=f\,(a)[/latex].
These three conditions are necessary for continuity of a function of two variables as well.
Definition
A function [latex]f\,(x,\ y)[/latex] is continuous at a point [latex](a,\ b)[/latex] in its domain if the following conditions are satisfied:
- [latex]f\,(a,\ b)[/latex] exists.
- [latex]\displaystyle{\lim_{(x,\ y)\to{(a,\ b)}}}f\,(x,\ y)[/latex] exists.
- [latex]\displaystyle{\lim_{(x,\ y)\to(a,\ b)}}f\,(x,\ y)=f\,(a,\ b)[/latex].
Example: Demonstrating Continuity for a Function of Two Variables
Show that the function [latex]f\,(x,\ y)=\frac{3x+2y}{x+y+1}[/latex] is continuous at point [latex](5,-3)[/latex].
Try it
Show that the function [latex]f\,(x,y)=\sqrt{26-2x^{2}-y^{2}}[/latex] is continuous at point [latex](2,-3)[/latex].
Continuity of a function of any number of variables can also be defined in terms of delta and epsilon. A function of two variables is continuous at a point [latex](x_0,y_0)[/latex] in its domain for every [latex]\epsilon>0[/latex] there exists a [latex]\delta>0[/latex] such that, whenever [latex]\sqrt{(x-x_0)^2+(y-y_0)^2}<\delta[/latex] it is true, [latex]|f(x,y)-f(a,b)|<\epsilon[/latex]. This definition can be combined with the formal definition (that is, the epsilon-delta definition) of continuity of a function of one variable to prove the following theorems:
the sum of continuous functions is continuous
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].
the product of continuous functions is continuous
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].
the composition of continuous functions is continuous
Let [latex]g[/latex] be a function of two variables from a domain [latex]D\subseteq\mathbb{R}^{2}[/latex] to a range [latex]R\subseteq\mathbb{R}[/latex]. Suppose [latex]g[/latex] is continuous at some point [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 that maps [latex]\mathbb{R}[/latex] to [latex]\mathbb{R}[/latex] such that [latex]z_0[/latex] is in the domain of [latex]f[/latex]. Last, assume [latex]f[/latex] is continuous at [latex]z_0[/latex]. Then [latex]f\circ{g}[/latex] is continuous at [latex](x_0,y_0)[/latex] as shown in the following figure.
Let’s now use the previous theorems to show continuity of functions in the following examples.
Example: More Examples of Continuity of a Function of Two Variables
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.
Try it
Show that the functions [latex]f\,(x,\ y)=2x^{2}y^{3}+3[/latex] and [latex]g\,(x,\ y)=(2x^{2}y^{3}+3)^{4}[/latex] are continuous everywhere.
Watch the following video to see the worked solution to the above Try It
Functions of Three or More Variables
The limit of a function of three or more variables occurs readily in applications. For example, suppose we have a function [latex]f\,(x,\ y,\ z)[/latex] that gives the temperature at a physical location [latex](x,\ y,\ z)[/latex] in three dimensions. Or perhaps a function [latex]g\,(x,\ y,\ z,\ t)[/latex] can indicate air pressure at a location [latex](x,\ y,\ z)[/latex] at time [latex]t[/latex]. How can we take a limit at a point in [latex]\mathbb{R}^{3}[/latex]? What does it mean to be continuous at a point in four dimensions?
The answers to these questions rely on extending the concept of a [latex]\delta[/latex] disk into more than two dimensions. Then, the ideas of the limit of a function of three or more variables and the continuity of a function of three or more variables are very similar to the definitions given earlier for a function of two variables.
Definition
Let [latex](x_0,\ y_0,\ z_0)[/latex] be a point in [latex]\mathbb{R}^{3}[/latex]. Then, a [latex]\delta[/latex] ball in three dimensions consists of all points in [latex]\mathbb{R}^{3}[/latex] lying at a distance of less than [latex]\delta[/latex] from [latex](x_0,\ y_0,\ z_0)[/latex] – that is,
To define a [latex]\delta[/latex] ball in higher dimensions, add additional terms under the radical to correspond to each additional dimension. For example, given a point [latex]P=(w_0,\ x_0,\ y_0,\ z_0)[/latex] in [latex]\mathbb{R}^{4}[/latex], a [latex]\delta[/latex] ball around [latex]P[/latex] can be described by
To show that a limit of a function of three variables exists at a point [latex](x_0,\ y_0,\ z_0)[/latex], it suffices to show that for any point in a [latex]\delta[/latex] ball centered at [latex](x_0,\ y_0,\ z_0)[/latex], the value of the function at that point is arbitrarily close to a fixed value (the limit value). All the limit laws for functions of two variables hold for functions of more than two variables as well.
Example: Finding the Limit of a Function of Three Variables
Find [latex]\displaystyle\lim_{(x,\ y,\ z)\to(4,\ 1,\ -3)}\frac{x^{2}y-3z}{2x+5y-z}.[/latex]
Try it
Find [latex]\displaystyle\lim_{(x,\ y,\ z)\to(4,\ -1,\ 3)}\sqrt{13-x^{2}-2y^{2}+z^{2}}[/latex].