I've come across an exercise in Royden & Fitzpatrick that's got me a bit confused.
It claims that if $c$ is the local minimizer for $f$ in $(a,b)$, then $$\underline{D} f(c) \leq 0 \leq \overline{D} f(c).$$
Here $\overline{D}f(x)=\lim\limits_{h \to 0}\left[\sup\limits_{0<|t|\leq h}\frac{f(x+t)-f(x)}{t}\right]$ is the upper derivative, and $\underline{D}f(x)=\lim\limits_{h \to 0}\left[\inf\limits_{0<|t|\leq h}\frac{f(x+t)-f(x)}{t}\right]$ is the lower derivative of f.
Since $c$ is the local minimizer, for a small neighborhood around $c$, $f(c+t)-f(c)>0$ for any $t\in(0,h]$, so I do not see how I can ever get the lower derivative to be negative. No matter what value $t$ takes, it seems that $\frac{f(x+t)-f(x)}{t}>0$.