Is $f$ has a fixed point $l$ does that mean there exists an interval such that $f((l-\epsilon,l+ \epsilon)) \subset (l - \delta, l + \delta)?$

Question

Suppose $f$ is continuous on $\mathbb{R}$ and $f$ has a fixed point. I am working on a problem, and I wondered if it was true that if we set $\epsilon > 0$ would there exists $\delta > 0$ such that $$f((l - \delta, l + \delta )) \subset (l-\epsilon, l + \epsilon)$$

If yes, how is the $\delta$ in relation with $\epsilon$, I think it would smaller than $\epsilon$. Any theorems or proofs on this?

Answer 1

Yes, because, since $f$ is continuous at $l$, there is a $\delta>0$ such that$$f\bigl((l-\delta,l+\delta)\bigr)\subset(f(l)-\varepsilon,f(l)+\varepsilon)=(l-\varepsilon,l+\varepsilon).$$