If $f:[a,b]\to\mathbb{R}$ is continous at $s$ where $s \in [a,b)$, then if $s<b$ can I find an $x \in [a,b]$ with $s<x<s+\delta$?
$\delta$ is $\delta$ of the epsilon-delta definition of continuity. I was able to prove it graphically but is there a way to show it analytically?
You've added in a comment that $a\le s<b$. Without loss of generality you can assume that $\delta \le b-s$. Simply take $x = s + \frac {\delta} 2$.