Consider the function $g:[0,1] \to \mathbb{R}$ defined by $g(x)=1 - \sqrt{0.5-x}$ when $x \leq 0.5$ and $g(x)=1$ otherwise.
What is $\lim_{h \to 0^{+}} \inf \{\frac{g(0.5+t)-g(0.5)}{t} : 0< \lvert t \rvert <h\}$? I think it is $0$ because the limit from the right hand side is $0$.
And what is $\lim_{h \to 0^{+}} \sup \{\frac{g(0.5+t)-g(0.5)}{t} : 0< \lvert t \rvert <h\}$? I think it is $\infty$ because the limit from the left hand side is $0$.
This function looks like this:
