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 f′ are never zero on I, then the derivative of f−1 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 g ◦ f are continuous on R, then g is continuous on R.
d) Suppose that I 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 f′(a) and g′(b) both exist and are nonzero, then (g ◦ f)−1(x) is differentiable at x = g(b), and ((g ◦ f)−1)′ (g(b)) =(f′(a) · g′(b))−1.
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