question archive Let f : R → R be additive

Subject:MathPrice: Bought3

Let f : R → R be additive. That is, f(x + y) = f(x) + f(y) for all x, y ∈ R.

In addition, assume there are M > 0 and a > 0 such that if x ∈ [−a, a], then |f(x)| ≤ M. Prove that f is uniformly continuous. In particular, prove that there is a real number m such that f(x) = mx for all x ∈ R.