Analysis, Part II: Continuous Functions on Metric Spaces

September 12th, 2017

Definition: A function $f: X\rightarrow Y$ is continuous at $x_0$ if for every $\epsilon > 0$, there exists a $\delta > 0$ so that:

$$ d(x, x_0) < \delta \implies d(f(x), f(x_0)) < \epsilon $$

Theorem

The following are equivalent:

We can generalize this statement for functions which are everywhere continuous:

Theorem

The following are equivalent:

This tells us that the topology definition of open sets is exactly the same as the definition on metric spaces (used for calculus). Without a metric, we still have the definition about open sets.

As expected, compositions of continuous functions are continuous; addition, substraction, multiplication, max, and min are all continuous.

Also as expected, we can take direct sumss of functions. $f\oplus g$ is continuous at a point iff $f,g$ are both continuous at $x_0$.

Continuity and Compactness

Continuous functions interact well with compact sets in the following way.

Theorem

The image of a compact set under a continuous function is compact.

So continuity preserves compactness. The following property of compact sets is important.

Proposition

Let $f$ be a continuous function on a compact set. Then, $f$ is bounded, and it attains its maximum (and minimum) somewhere within the compact set.

A more useful generalization of continuity is uniform continuity – the key in this definition is that it does NOT depend on $x$.

Definition: $f$ is uniformly continuous if for every $\epsilon > 0$ there exists a $\delta > 0$ so that:

$$ d(x, y) < \delta \implies d(f(x), f(y)) < \epsilon $$

For any choice of $x,y$.

And on a compact domain these two notions are equivalent!

Theorem

Let $X$ be compact. Then $f: X \rightarrow Y$ is continuous iff it is uniformly continuous.

Connectedness

Definition: $X$ is disconnected if there exist disjoint non-empty open sets $V,W$ so that $X=V\cup W$. $X$ is connected if it is nonempty and not disconnected.

Note that in this definition, $V, W$ are relatively open with respect to $X$. If there exists a larger space $X \subset Y$, To say that $V$ is open relatively with respect to $X$ is to say that $V = X \cap V’$ for some open set $V’ \subset Y$. So this means that the definition of connectedness is intrinsic and does not depend on the ambient space.

Corollary

$X$ is connected iff the only clopen sets are $X$ itself and the empty set.

On the real line, the connected sets are precisely intervals.

Finally, continuity preserves connectedness just like it does compactness.

Theorem

Let $f: X\rightarrow Y$ be a continuous map. Let $E \subset X$ be connected. Then $f(E)$ is connected.

The simple reason why is from the definition; we can directly map back a disjoint union of open sets to a disjoint union of open sets.

As a direct corollary, we get the Intermediate Value Theorem.

Corollary

Let $f: X \rightarrow \mathbb{R}$ be continuous. Let $E$ be connected in $X$, with $a,b \in E$. If $y\in (f(a), f(b))$, then there exists $c\in E$ so that $f(c)=y$.

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