A property of continuous functions

Question

A function $f : [a,b] \to [a,b]$ is continuous for all $x \in [a,b].$ Prove that there exists a $c\in [a,b]$ such that $f(c) = c.$

score 2 · Answer 1 · answered Oct 04 '13 at 12:04

2

Hint: Use intermediate value theorem on $g(x) = f(x) - x$

answered Oct 04 '13 at 12:04

score 1 · Answer 2 · edited Jul 01 '14 at 00:47

1

Hint: $f:[a,b]\to \mathbb{R}$ be continuous, such that $f(a)f(b)<0$. Then there exists $c\in(a,b)$ such that $f(c)=0$.

edited Jul 01 '14 at 00:47

answered Oct 04 '13 at 12:04

Myshkin

2 Answers2