Paper 1, Section II, E

Analysis I | Part IA, 2013

(a) Let f:[a,b]Rf:[a, b] \rightarrow \mathbb{R}. Suppose that for every sequence (xn)\left(x_{n}\right) in [a,b][a, b] with limit y[a,b]y \in[a, b], the sequence (f(xn))\left(f\left(x_{n}\right)\right) converges to f(y)f(y). Show that ff is continuous at yy.

(b) State the Intermediate Value Theorem.

Let f:[a,b]Rf:[a, b] \rightarrow \mathbb{R} be a function with f(a)=c<f(b)=df(a)=c<f(b)=d. We say ff is injective if for all x,y[a,b]x, y \in[a, b] with xyx \neq y, we have f(x)f(y)f(x) \neq f(y). We say ff is strictly increasing if for all x,yx, y with x<yx<y, we have f(x)<f(y)f(x)<f(y).

(i) Suppose ff is strictly increasing. Show that it is injective, and that if f(x)<f(y)f(x)<f(y) then x<y.x<y .

(ii) Suppose ff is continuous and injective. Show that if a<x<ba<x<b then c<f(x)<dc<f(x)<d. Deduce that ff is strictly increasing.

(iii) Suppose ff is strictly increasing, and that for every y[c,d]y \in[c, d] there exists x[a,b]x \in[a, b] with f(x)=yf(x)=y. Show that ff is continuous at bb. Deduce that ff is continuous on [a,b][a, b].

Typos? Please submit corrections to this page on GitHub.