In the textbook, FONC states:
With the proof:
I don't understand the following:
Why should that condition hold simply because $\nabla f$ is continuous near $x^*$? I tried to apply the $\epsilon-\delta$ definition of continuity to produce such a scalar $T > 0$, but I could not proceed.
In the next line, why do we pick a new variable $\bar t \in (0, T]$? The interval $(0, T]$ seems to be quite mysterious to me. (Solved)