So I'm new to uniform continuity and this is just an exercise that was scribbled on the board in my real analysis class. But it's is a tricky question for me:
if $f:R\to R$ is periodic with period P (so $f(x+P)=f(x)$) and $f$ is continuous on $(0,p+a)$ for some $\epsilon >0$, then $f$ is uniformly continuous
how can we go about proving this?
There is a well-known theorem about uniform continuity of a real-valued function -- which does most of the work in answering this question.
Theorem: A continuous function with a compact domain is uniformly continuous.
So the restriction of the given function $f(x)$ to the interval $[0,P]$ is uniformly continuous. Now since $f(x)$ is periodic we can always relate its "rate of change" $\Delta_\varepsilon(x) = \displaystyle \frac{f(x+\varepsilon)-f(x)}{\varepsilon}$ to $\Delta_\varepsilon(y)$ for some $y \in [0,P]$.