 Solution

With the strict monotonicity assumed at the start of the question, parts (a) and (c) give a strictly increasing inverse functionIts image of a function is exactly the interval between its endpoint values. Apply part (c) to $g$, with domain $[f(a),f(b)]$, to conclude that $g$ is a continuous function, including at both endpoints.

If the word “increasing” in this subpart were interpreted as merely nondecreasing, an inverse function need not exist: a constant function is a counterexample. The preceding strict hypothesis supplies the required injectivity.