You made a mistake. The derivative is bounded.
Prove that is either uniform continuous or not.
I claim that this function is not, since its derivative is not bounded. But how do I show that using the definition?
Proof so far:
Let , for every , I need to find implies
What kind of x and y should I pick?
Theorem: a continuous periodic function is uniformly continuous.
Indeed, it is uniformly continuous on two periods by Heine's Theorem (a continuous function defined on a compact is uniformly continuous), and by periodicity this implies the uniform continuity on (choose so that, if , then and lie in the same interval of length and the situation matches a similar situation inside ).
And if you don't know this theorem by Heine, you can write .