Suppose that f,g : [a, b] → R. Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones.
a) If f = g2 and / is differentiable on [a, b], then g is differentiable on (a, b).
b) If f is differentiable on [a, b], then f is uniformly continuous on [a, b].
c) If f is differentiable on (a, b) and f(a) = f(b) = 0, then f is uniformly continuous on [a, b].
d) If f is differentiable on (a, b] and f(x)/(x - a) → 1 as x → a+, then f is uniformly continuous on [a, b].