I think it is, here is my proof:

Suppose that $\displaystyle f: \mathbb {R} \mapsto \mathbb {R} $ is a t-periodic function. We will show that f is uniformly continuous.

Let $\displaystyle \epsilon > 0 $, find $\displaystyle \delta > t $ such that $\displaystyle x \in \mathbb {R} $ with $\displaystyle y = x + t $, then $\displaystyle |x-y| = | x - x + t | = |-t| = t < \delta $

We then have $\displaystyle |f(x) - f(y) | = |f(x) - f(x + t)| = |f(x)-f(x)| =0 < \epsilon $, thus proves the claim.

Is this right? Thank you.