Is function $f$ also uniformly continuous?

Question

I've been thinking on the following problem lately:

Let $(X,d)$ be a metric space and $f_1,f_2,...,f_n: X \rightarrow \mathbb{R}$ and

$f(x) = \max\{f_1(x),f_2(x),...,f_n(x) \}$,$x\in X$

If the functions $f_1,f_2,...,f_n$ are uniformly continuous, is $f$ also uniformly continuous?

I feel like it is(since it's obvious), just having difficulties proving it.

Can I get any help starting my proof?

Thank you!

Answer 1

In choosing an $x,y$ with $|x-y|<\delta$, if the same $f_n$ is realizes the maximum for both $x$ and $y$, then we're done by the uniform continuity of $f_n$, otherwise, if $f_m(x)$ is the max for $x$ and $f_n(y)$ is max for $y$, then $f_m(t)-f_n(t)$ is zero for some $t$ between $x$ and $y$ by intermediate value. So then use $|x-y|<|x-t|+|t-y|$ by triangle inequality.