Continuous mapping on a compact metric space is uniformly continuous

I am struggling with this question:

Prove or give a counterexample: If $f : X \to Y$ is a continuous mapping from a compact metric space $X$, then $f$ is uniformly continuous on $X$.

Thanks for your help in advance.

Answer

The answer is yes, if $f$ is continuous on a compact space then it is uniformly continuous:

Let $f: X \to Y$ be continuous, let $\varepsilon > 0$ and let $X$ be a compact metric space. Because $f$ is continuous, for every $x$ in $X$ you can find a $\delta_x$ such that $f(B(\delta_x, x)) \subset B({\varepsilon\over 2}, f(x))$. The balls $\{B(\delta_x, x)\}_{x \in X}$ form an open cover of $X$. So do the balls $\left\{B \left(\frac{\delta_x}{2}, x\right)\right\}_{x \in X}$. Since $X$ is compact you can find a finite subcover $\left\{B \left( \frac{\delta_{x_i}}{2}, x_i \right) \right\}_{i=1}^n$. (You will see in a second why we are choosing the radii to be half only.)

Now let $\delta_{x_i}’ = {\delta_{x_i}\over 2}$.

You want to choose a distance $\delta$ such that for any two $x,y$ they lie in the same $B(\delta_{x_i}’, x_i)$ if their distance is less than $\delta$.

How do you do that?

Note that now that you have finitely many $\delta_{x_i}’$ you can take the minimum over all of them: $\min_i \delta_{x_i}’$. Consider two points $x$ and $y$. Surely $x$ lies in one of the $B(\delta_{x_i}’, x_i) $ since they cover the whole space and hence $x$ also lies in $B(\delta_{x_i}’, x_i)$ for some $i$.

Now we want $y$ to also lie in $B(\delta_{x_i}’, x_i)$. And this is where it comes in handy that we chose a subcover with radii divided by two:

If you pick $\delta : = \min_i \delta_{x_i}’$ (i.e. $\delta = \frac{\delta_{x_i}}{2}$ for some $i$) then $y$ will also lie in $B(\delta_{x_i}, x_i)$:

$d(x_i, y) \leq d(x_i, x) + d(x,y) < \frac{\delta_{x_i}}{2} + \min_k \delta_{x_k} \leq \frac{\delta_{x_i}}{2} + \frac{\delta_{x_i}}{2} = \delta_{x_i}$.

Hope this helps.

Attribution
Source : Link , Question Author : the code , Answer Author : AJY

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