A measure of non-differentiability Consider $f(x) = x^2$ and $g(x) = |x|$. Both graphs have an upward open graph, but $g(x) = |x|$ is "sharper". Is there a way to measure this sharpness? 

I think the notion which fits the bill is Hölder continuity.
Probably your example is not the most illustrative one (since $f$ is infinitely differentiable at 0). A function $f$ is $C^{k,\alpha}$ Hölder continuous (with $k\in\mathbb{N}$ and $\alpha\in[0,1]$) if it is $k$-times differentiable and for its $k$th derivative $f^{(k)}$ there exists a constant $C$ such that
$$
\sup_{x\neq y}\frac{|f(x) - f(y)|}{|x-y|^\alpha}\leq C.
$$ 
Anyway, $g(x) = |x|$ is in the space $C^{0,1}$, i.e. not differentiable but Lipschitz continuous. Sharper cusps, like $h(x) = \sqrt{|x|}$ have lower Hölder regularity ($C^{0,1/2}$ in this case).
