= Solution
{c}
With the strict <monotonicity> assumed at the start of the question, parts (a) and (c) give a <strictly increasing> <inverse function>
$$
g:[f(a),f(b)]\longrightarrow[a,b],\qquad g(f(a))=a,\quad g(f(b))=b.
$$
Its <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 \b[$g$ 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>.
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