continuity

Section 4.2 continuity

Being able to compute limits by evaluation is such a nice condition that we give it a name:

Definition 4.2.1.

We say \(f:A\to V\) is continuous at \(c\in A\) if either \(c\) is not a limit point for \(A\text{,}\) or

\begin{equation*} \displaystyle\lim_{x\to c}f(x)=f(c)\ \ . \end{equation*}

Example 4.2.2.

A polynomial is continuous at each point of its domain.

Proposition 4.2.3. sequential characterization of continuity at a point.

\(f:A\to V\) is continuous at \(c\) if and only if, for any sequence \((x_n)_{n\in\mathbb{N}}\) with

\(\displaystyle \forall n, x_n\in A\)
\(\displaystyle x_n\to c\)

we have \(f(x_n)\to f(c)\text{.}\)

Checkpoint 4.2.4.

Observe that the sequential characterization of continuity at a point has a slightly different hypothesis than the sequential characterization of function limits. Explain this.

Checkpoint 4.2.5.

If we think of a sequence as a function \(X:\mathbb{N}\to\mathbb{R}\text{,}\) and regard \(\mathbb{N}\subseteq\mathbb{R}\text{,}\) any sequence is continuous at each point of \(\mathbb{N}\text{.}\)

Definition 4.2.6.

If a function \(f\) is continuous at \(c\) for all \(c\in B\subseteq A\text{,}\) we say \(f\) is continuous on \(B\).

If a function \(f\) is continuous on its domain, we say \(f\) is continuous.

Checkpoint 4.2.7.

Show that the following metric characterization of continuity holds: \(f\) is continuous at \(c\) if either \(c\) is not a limit point for \(A\text{,}\) or

\begin{equation*} \forall\epsilon\gt 0,\ \exists\delta\gt 0: d(x,c)\lt \delta \Rightarrow d(f(x),f(c))\ \ . \end{equation*}

Checkpoint 4.2.8.

In Checkpoint 4.2.7, upon what is \(\delta\) allowed to depend?

Checkpoint 4.2.9.

Here's a sequential characterization of continuity on a domain:

\(f\) is continuous if it commutes with the taking of sequence limits.

Formalize and prove this statement.

Proposition 4.2.10. metric characterization of continuity on a domain.

\(f:A\to V\) is continuous if and only if, for every \(x\in A\) and every \(\epsilon\gt 0\text{,}\) there is \(\delta\gt 0\) so that \(d(x,y)\lt \delta\) guarantees \(d(f(x),f(y))\lt \epsilon\text{.}\)

Proof.

This isn't a new fact; it's just recording what continuity on a domain means for future reference.

Theorem 4.2.11.

Let \(A\subset U\) be a subset of a metric space, \(V\) and \(W\) metric spaces, \(f:A\to V\text{,}\) and \(g:f(A)\to W\text{.}\) If \(f\) is continuous and \(g\) is continuous, then \(g\circ f\) is continuous.

Proof.

There are two proofs: a metric proof, involving \(\epsilon\)s and \(\delta\)s, and a sequential proof, using Proposition 4.2.3.

Shortly, we'll give a topological proof as well.