Preservation of Compact Sets (confused about counterexample).

Question

This definition for the preservation of compact sets is taken from Abott 2001:

Let $f : A \to \Bbb R$ be continuous on $A$. If $K \subseteq A$ is compact, then $f(K)$ is compact as well.

I feel like I understand the general proof pretty well, but I can't seem to get my head around a particular counter example I came up with.

Consider the function $f: [0,1] \to \mathbb R$ where $f(x) = \dfrac{1}{x}$ .

Now I've restricted the domain to a compact set (the closed interval), but the range runs from $[1, \infty)$.

So it seems to be I've mapped from a compact set with a continuous function, but the image isn't compact (because it's unbounded).

I'm clearly missing something—help?

Thanks!

Answer 1

$f(0) = 1/0$ is undefined.

So $f:[0,1] \not \rightarrow \mathbb R$

$f:(0,1] \rightarrow [1, \infty)$.

Which is fine as neither $(0,1]$ nor $[1,\infty)$ are compact.