Continuous function.

Question

Suppose that function $f:\mathbb[0,1]\to\mathbb{R}$ is continuous and that $f(x)>2$ if $0\leq x< 1$. Is it necessarily the case that $f(1)>2$?

So I'm thinking about setting $f(1)\leq2$ and try to prove that $f$ is continuous at $x=1$. So it means that when ${x_n}\to1$, $f(x_n)\to f(1)$ but that's all I can think of.

score 1 · Answer 1 · answered Oct 23 '13 at 08:32

1

Consider $f(x) = 3-x{}{}{}{}{}$.

answered Oct 23 '13 at 08:32

mrf

43,639

score 1 · Answer 2 · answered Oct 23 '13 at 08:33

1

No. Take $$f(x) = -x+3$$ In general, you will only get that $f(1) \geq 2$

answered Oct 23 '13 at 08:33

Prahlad Vaidyanathan

31,554

Continuous function.

2 Answers2