Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones.
a) Suppose that I ⊂ R is nonempty. If f : I → R is 1-1 and continuous, then f is strictly monotone on I.
b) Suppose that I is an open interval which contains 0 and that f: I → R is 1-1 and differentiable. If f and fʹ are never zero on I, then the derivative of f-l has at least one root in f(I); that is, there is an a ∈ I such that (f-1)'(a) = 0.
c) Suppose that f and g are 1-1 on R. If f and go f are continuous on R, then g is continuous on R.
d) Suppose that f is an open interval and that a ∈ I. Suppose further that f : I → R and g : f(I) → R are both 1-1 and continuous and that b := f(a). If fʹ(a) and g'(b) both exist and are nonzero, then (g o f)-1 (x) is differentiable at x = g(b), and ((g o f)-l)'(g(b)) = (fʹ(a) ∙ gʹ(b)-1].