Fixed value property of $[0,1]$

Question

I know every continuous function from $ [0,1]$ to $[0,1]$ has fixed point. I proved it using Intermediate value theorem. But I can not see why every continuous function from $[0,1]$ to $\mathbb{R}$ has fixed point.

Is this result even true?

Thanks a lot.

Answer 1

No. $f(x)=x+1$ is a continuous function from $[0,1]$ to $\mathbb R$.