Real Analysis: Uniformly continuous

[annastm]

Please help me with this one:

Suppose f is a continuous real valued function on R - real #s and that 0<f(x)<3 for every x in R. Let F(x) = integral from 0 to x of f(t) dt.

a) Show that F is uniformly continuous on R
b) Suppose the continuity assumption is left out, but the function f is still assumed to have 0<f(x)<3 for all x and to be integrable on every bounded interval. Must the conclusion of uniform continuity of F on R still be true? Prove or give counterexample.

Thank You for your help

 

[matt grime]

By the (second?) Fundamental Theorem of Calculus, F, in part a), is a differentiable function with (uniformly) bounded derivative. It is a standard exercise to show this is uniformly continuous, so what do you think you need to do now you've got this (slightly) different way of thinking about it?