Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics II / 4 / c

If $f$ is a continuous function, the intermediate value theorem ensures that every value between $f(a)$ and $f(b)$ is attained. Monotonicity excludes values outside that range, so $f([a,b])=[f(a),f(b)]$.

Conversely, suppose this equality of sets holds. At an interior point, if $f^{-}(x_0)<f(x_0)$, any $y$ strictly between these two numbers lies in $[f(a),f(b)]$. Yet inputs below $x_0$ have values at most $f^{-}(x_0)$ and inputs at or above $x_0$ have values at least $f(x_0)$, so no input attains $y$. This contradicts surjectivity. The same argument excludes $f(x_0)<f^{+}(x_0)$, so part (b) gives continuity at every interior point.

At $a$, a strict gap $f(a)<f^{+}(a)$ likewise leaves every value between them unattained; hence $f^{+}(a)=f(a)$, giving right continuity. At $b$, the corresponding argument gives $f^{-}(b)=f(b)$ and left continuity. These are the appropriate endpoint notions on $[a,b]$. Hence