Paper 1, Section II, F

Analysis I | Part IA, 2021

(a) State the intermediate value theorem. Show that if f:RRf: \mathbb{R} \rightarrow \mathbb{R} is a continuous bijection and x1<x2<x3x_{1}<x_{2}<x_{3} then either f(x1)<f(x2)<f(x3)f\left(x_{1}\right)<f\left(x_{2}\right)<f\left(x_{3}\right) or f(x1)>f(x2)>f(x3)f\left(x_{1}\right)>f\left(x_{2}\right)>f\left(x_{3}\right). Deduce that ff is either strictly increasing or strictly decreasing.

(b) Let f:RRf: \mathbb{R} \rightarrow \mathbb{R} and g:RRg: \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 ff and gg are continuous then fgf \circ g is continuous.

(ii) If gg is strictly increasing and fgf \circ g is continuous then ff is continuous.

(iii) If ff is continuous and a bijection then f1f^{-1} is continuous.

(iv) If ff is differentiable and a bijection then f1f^{-1} is differentiable.

Typos? Please submit corrections to this page on GitHub.