Paper 1, Section II, F

(a) State the intermediate value theorem. Show that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous bijection and $x_{1}<x_{2}<x_{3}$ then either $f\left(x_{1}\right)<f\left(x_{2}\right)<f\left(x_{3}\right)$ or $f\left(x_{1}\right)>f\left(x_{2}\right)>f\left(x_{3}\right)$. Deduce that $f$ is either strictly increasing or strictly decreasing.

(b) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be functions. Which of the following statements are true, and which can be false? Give a proof or counterexample as appropriate.

(i) If $f$ and $g$ are continuous then $f \circ g$ is continuous.

(ii) If $g$ is strictly increasing and $f \circ g$ is continuous then $f$ is continuous.

(iii) If $f$ is continuous and a bijection then $f^{-1}$ is continuous.

(iv) If $f$ is differentiable and a bijection then $f^{-1}$ is differentiable.

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