Limits and continuity

The concept of limits and continuity of real-valued functions in several variables.

The concepts of limits and continuity form the foundation of differential calculus. In this lecture, we will develop these concepts in the context of real-valued functions of several variables.

Notations

Much of what we will discuss here apply equally well to 2-dimensional, 3-dimensional spaces, or even higher-dimensional spaces. Therefore, we will directly talk about functions of $n$ variables defined on subsets of $\mathbb{R}^n$.

Throughout our discussions, we will use the boldface symbol for ordered list of $n$ variables, i.e., we will write \[ \mathbf{x} = (x_1,\dots,x_n). \]

We also make frequent use of the distance function for $n$-dimensional spaces, which is given by the familiar distance formula (Pythagorean Theorem).

For any two points $\mathbf{x} = (x_1,\dots,x_n)$ and $\mathbf{y} = (y_1,\dots,y_n)$ in $\mathbb{R}^n$, we define \[ d(\mathbf{x},\mathbf{y}) = \sqrt{ (x_1-y_1)^2 + \cdots + (x_n - y_n)^n }. \]

Limits

With minor modifications, the concept of limits can be directly generalized to functions of several variables.

Suppose $f$ is a real-valued function of $n$ variables. We say the limit of $f$ as $\mathbf{x}$ approaches $\mathbf{z}$ is $L$ and write \[ \lim_{\mathbf{x} \to \mathbf{z}} f(\mathbf{x}) = L \] if for any real number $\epsilon > 0$, there exists a real number $\delta >0$ s.t. \[ 0 < d(\mathbf{x},\mathbf{z}) < \delta \;\text{implies}\; |f(\mathbf{x}) - L| < \epsilon. \]

Here, we are assuming $f$ is defined in an "neighborhood" around $\mathbf{z}$, but it may or may not be defined at $\mathbf{z}$.

Of course, if no such real number $L$ exists, we say $\lim_{\mathbf{x} \to \mathbf{z}} f(\mathbf{x})$ does not exist (or undefined).

When the limit exists, familiar "laws" of limits we learned earlier remain valid.

When limits fail to exist

It is important to notice that for $n > 1$, limits of a real-valued function in $n$ variables may fail to exist for some new reasons.

For $n > 1$, there are actually infinitely many directions from which we can approach a point $\mathbf{z}$ in $\mathbb{R}^n$. If the limit exists, it must be independent from the direction.

Show the limit \[ \lim_{(x,y) \to (0,0)} \frac{ xy }{x^2 + y^2} \] does not exist.

It may be helpful to plot the graph of this function near the origin $(0,0)$ to see exactly what the problem is.

Continuity

Just like in the calculus of functions in one variable, we can define the concept of continuity for functions of several variables in terms of limits.

For a real-valued function $f(\mathbf{x})$ of $n$ variables, we say $f$ is continuous at $\mathbf{z}$ if
  1. $f(\mathbf{z})$ is defined, i.e., $z$ is in the domain of $f$;
  2. $\lim_{\mathbf{x} \to \mathbf{z}} f(\mathbf{x})$ exists (and is finite); and
  3. $\lim_{\mathbf{x} \to \mathbf{z}} f(\mathbf{x}) = f(\mathbf{z})$.

In other words, a function being continuous at a point means its function value and limit at this point are both defined and they are the same.

This concept mostly agrees our intuition: at a point where a function is continuous, the graph of the function has no gaps and holes.

Properties of continuous functions

As expected, the

  • sums,
  • differences,
  • products, and
  • compositions

of continuous functions in several variables are still continuous.

Notice that we did not include quotients of continuous functions in several variables in this list. In general, the quotient of two continuous functions may not be continuous.

What could go wrong?

Of course, for function compositions, we need to make sure they actually make sense. That is, if $g$ is a real-valued functions in $n$ variables, the composition $f(g(x_1,\ldots,x_n))$ only makes sense if $f$ is a function in one variable.