TRUE/FALSE: Function f is integrable on [a, b] provided that f is uniformly continuous on [a, b]

Question

Statement: "TRUE/FALSE: Function f is integrable on [a, b] provided that f is uniformly continuous on [a, b]"

I'm not sure if I understand what "provided that" means. Does this mean that the proof would work either way? Is it the same as saying "A function f on [a, b] which is uniformly continuous is Riemann integrable on [a, b]."

Answer 1

Uniformly continuous implies bounded implies (with continuity) integrable. The reverse is not true. Discontinuous functions may be integrable.