f is cont on [a, b] and g is : g(a) = f(a) and, for a < x <= b, g(x) is max value of f in the [a, x]. Show g is continuous on [a, b]

Question

The highlighted parts are not clear for me. Can you eplain me, why do we consider two cases and what is exactly going on in both of them.

Edited:

Especially it is not clear, why do we consider $p_1 <= c - \sigma^{`}$ and $p_1 > c - \sigma^{`}$

Answer 1

There has to be a difference of treatment of the difference $|g(x)-g(c)|$ whether $f$ takes its maximum on $[a,c+\delta']\cap [a,b]$ inside $[c-\delta',c+\delta']$ or outside :

• if it is outside (which means before), then $g$ is constant on $[c-\delta',c+\delta']$, therefore continuous at $c$ ;
• if it is inside, then you use the continuity of $f$ to show that $g(x)$ can't be far from $g(c)$ inside $[c-\delta',c+\delta']$ (because it's not far from $f(c)$) ; again, it means $g$ is continuous at $c$.