Continuity of Functions of Several Variables

# Continuity of Functions of Several Variables

Recall that a function of a single variable $y = f(x)$ is continuous at $c \in D(f)$ if $\lim_{x \to c} f(x) = f(c)$. We will now extend the concept of continuity of a function of a single variable to a function of several variables.

 Definition: A two variable real-valued function $z = f(x, y)$ is said to be Continuous at $(a, b) \in D(f)$ if $\lim_{(x,y) \to (a,b)} f(x,y) = f(a,b)$. If $f$ is not continuous at $(a,b)$, then we say that $f$ is Discontinuous at $(a, b)$. We say that $f$ is a Continuous Two Variable Function if $f$ is continuous for all $(a, b) \in D(f)$.

Geometrically, a two variable function $z = f(x, y)$ is continuous if the graph of $f$ does not contain any holes, breaks, jumps, or asymptotes.

In determining discontinuities of a two variable real-valued function, we need to only consider points of the function that would normally cause discontinuities in single variable real-valued functions. For example, consider the following two variable function:

(1)
\begin{align} f(x, y) = \frac{x^2 + y^2}{1 + x} \end{align}

We note that both the numerator and denominator is continuous for all $x, y \in \mathbb{R}$ as they're both polynomial functions. However, notice that if $x = -1$ then $f$ is not defined. Therefore, the points $(-1, y) \in \mathbb{R}^2$ where $y \in \mathbb{R}$ are the only discontinuities to $f$, and so $f$ is continuous elsewhere.

Of course, we can extend to concept of continuity to functions of three or more variables.

 Definition: A three variable real-valued function $w = f(x, y, z)$ is said to be Continuous at $(a, b, c) \in D(f)$ if $\lim_{(x,y,z) \to (a,b,c)} f(x,y,z) = f(a,b,c)$. An $n$ variable real-valued function $z = f(x_1, x_2, ..., x_n)$ is said to be Continuous at $(x_1, x_2, ..., x_n) \in D(f)$ if $\lim_{(x_1, x_2, ..., x_n) \to (a_1, a_2, ..., a_n)} f(x_1, x_2, ..., x_n) = f(a_1, a_2, ..., a_n)$.

For example, consider the following three variable real-valued function:

(2)
\begin{align} f(x,y,z) = \frac{x^2 + y^2 + z^2}{\sqrt{x + y}} \end{align}

We see that both the numerator and denominator are continuous on their respective domains, so we will look at where $f$ is undefined. If $x + y < 0$, then $f$ is not defined, and so $f$ is discontinuous for $y > -x$.

Let's look at some more examples.

## Example 1

Determine any points of discontinuity for the function $f(x, y) = x^2y + 2xy + y^2 - 4y + 3$.

We note that $f$ represents a polynomial, and we know that polynomials are defined for all real numbers. Therefore, $f$ is discontinuous nowhere, i.e, $f$ is continuous on all of $\mathbb{R}^2$.

To show this, let $(a, b) \in \mathbb{R}^2$. Then:

(3)
\begin{align} \quad \lim_{(x,y) \to (a,b)} f(x,y) = \lim_{(x,y) \to (a,b)}x^2y + 2xy + y^2 - 4y + 3 = a^2b + 2ab + b^2 - 4b + 3 = f(a, b) \end{align}

So $f$ is continuous for all $(a, b) \in \mathbb{R}^2$.

## Example 2

Determine any points of discontinuity for the function $f(x, y) = \tan x + \sin y$.

We note that $\tan x$ is undefined if $x = (2k-1)\frac{\pi}{2}$ where $k \in \mathbb{Z}$, and so $f$ is discontinuous for $(x,y) = \left ([2k-1]\frac{\pi}{2}, y \right )$ where $k \in \mathbb{Z}$. The graph of $f(x,y) = \tan x + \sin y$ is given below.

## Example 3

Determine any points of discontinuity for the function $f(x,y) = \left\{\begin{matrix} x^2 + y^2& \mathrm{if} \: (x,y) \neq (1, 1) \\ 3 & \mathrm{if} (x, y) = (1, 1) \end{matrix}\right.$.

Clearly $f$ is defined for all $(x, y) \in \mathbb{R}^2$, however, $f$ is not continuous at $(1, 1)$. Notice that $\lim_{(x, y) \to (1,1)} f(x,y) = 2 \neq 3 = f(1, 1)$.

## Example 4

Determine any points of discontinuity for the function $f(x, y, z) = \sqrt{e^x + \frac{2e^y}{\sin x} + \frac{3}{z}}$.

Notice that if $\sin x = 0$ or $z = 0$ then $f$ is undefined. We note that $\sin x = 0$ if $x = k\pi$ where $k \in \mathbb{Z}$, and so $\{ (x, y, z) : x = k\pi \: \mathrm{or} \: z = 0 \}$ is the set of points of discontinuity to $f$.