Decide which of the following statements are true and which

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.

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