Show that if $f$ is defined on $(a, b)$ and $c\in(a, b)$ is a local minimizer for f, then $\underline{D}f(c) \leq 0 \leq \overline{D}f(c)$.
Proof:
There exists $\delta > 0$ such that $f(c) < f(c-h)$ for $0
Why can we flip f(c) and f(c-h)? Shouldn't it still be $\underline{D}f(c) = \underline{lim}_{h\rightarrow0}\frac{f(c-h)-f(c)}{h} \geq0$ which wouldn't help?
Likewise, There exists $\delta' > 0$ such that $f(c) < f(c+h)$ for $0
Then, $\underline{D}f(c) \leq 0 \leq \overline{D}f(c)$.