Is a periodic function differentiable? uniformly continuous? Let $f : \mathbb R 
\rightarrow \mathbb R$ be a function such that $f(x + 1) = f(x)$ for all $x \in \mathbb R.$ Which of the
following statement(s) is/are true?
(A) $f$ is bounded.
(B) $f$ is bounded if it is continuous.
(C) $f$ is differentiable if it is continuous.
(D) $f$ is uniformly continuous if it is continuous.
It is a periodic function with period 1. I don't know how to proceed further... 
Any Hints will be appreciated... 

A. It doesn't have to be bounded: Does a periodic function have to be bounded?
B. If it is continuous, then the image $f(\mathbb R) = f([0,1])$ is compact as $[0,1]$ is compact and hence must be bounded. If results about compact sets is not familiar to you, use the extreme value theorem.
What are your thoughts on C and D?
