Counterexample of "Continuous Function on Compact Space is Uniformly Continuous"?

Question

I can't see why the following could not be a counterexample.

Let $f:[-1;1]\to\mathbb{R}$ defined as $f(x) = \sqrt[3]{x}$. It's a continuous function that goes from a compact set to $\mathbb{R}$ but it's not uniformly continuous.

I really appreciate any kind of help.

Answer 1

A continuous function on compact space is uniformly continuous.

Then a function $f:X\to Y$ is uniformly continuous on $X$ if and only if:

$$(\forall\epsilon>0)(\exists\delta>0)(\forall x_1,x_2\in X): |x_1-x_2|<\delta \implies |f(x_1)-f(x_2)|<\epsilon.$$

We will show that $f:[-1;1]\to\mathbb{R}$ defined by the rule $f(x) = \sqrt[3]{x}$ is uniformly continuous.

Let $\epsilon>0$. Use the inequality $|\sqrt[3]{x} - \sqrt[3]{y}\big| \leq \sqrt[3]{|x-y|}$, to continue the proof.