7. Suppose that I is continuous on a closed, bounded interval [a, b].

(a) If I is differentiable on

(a,

b) and I'(x) 2: co> 0 for all x E

(a, b), prove that

(I-I)' exists and is bounded on (I(a), I(b)).

(b) If I is continuously differentiable on

(a,

b) and I'(xo) =I- 0 for some Xo E

(a, b), prove that there are intervals I and J such that I is 1-1 from I onto J and 1-1 is continuously differentiable on J.