Pointwise continuity and supremum/infimum.

Question

Does continuity of a function $f:[a,b]\to\mathbb{R}$ at a point $c$, $a<c<b$ imply that it achieves it supremum/infimum around that point?

Let's say we for an $\varepsilon$ chose a $\delta$ such that $d(x,c)<\delta$, does this imply that the supremum on that interval is attained?

I know that this is the case when the interval is compact, but in this case it's open $(c-\delta,c+\delta)$ and, I surmise, not compact.

Answer 1

No this can't be.

Take $f(x) = x$ on a unit interval $[0,1]$. Then if we look at the open set around $c = 0.1$, so for example $(0.09, 0.11)$ one can see that the supremum of $f$ on that interval is the same as the supremum of that interval which is $0.11$. Yet $0.11 \not\in (0.09, 0.11)$.

Answer 2

Yes, but not necessarily on that interval. It may be achieved on the boundary, hence compactness is needed as you mention.

Consider $f(x)=x$ on $(a,b)$.