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 is continuous on [a, b] and J := f([a, b]), then J is a closed, bounded interval.
b) If f and g are continuous on [a, b], if f(a) < g(a) and f(b) > g(b), then there is a c ∊ [a, b] such that f(c) = g(c).
c) Suppose that f and g are defined and finite valued on an open interval I which contains a, that f is continuous at a, and that f(a) ≠ 0. Then g is continuous at a if and only if fg is continuous at a.
d) Suppose that f' and g are defined and finite valued on R. If f and g o f are continuous on R, then g is continuous on R.