Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 1 / 9F / b

True. If $x_n\to x$, continuity of $g$ gives $g(x_n)\to g(x)$, and continuity of $f$ then givesEquivalently, this is the composition of continuous functions theorem.

This can be false because a strictly increasing $g$ need not be surjective. TakeThen $g$ is strictly increasing and $f\circ g=0$ is continuous, while $f$ is discontinuous at zero.

True. Part (a) shows that $f$ is strictly monotone. Suppose it is increasing and let $y_0=f(x_0)$. Given $ϵ>0$,Choose $\delta >0$ smaller than both distances from $y_0$ to the two outer values. If $|y-y_0|<\delta$, monotonicity forcesThus $f^{-1}$ is continuous at $y_0$. The decreasing case is identical with inequalities reversed.

This can be false. The functionis a differentiable bijection of $\mathbb{R}$, buthas an unbounded difference quotient at zero and is not differentiable there. The inverse function theorem requires the additional local condition $f^{\prime }(x)\ne 0$.