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? 

This may be somewhat above your pay grade, but a "measure" of a discontinuity of a function at a point may be seen in a Fourier transform of that function.  For example consider the function
$$f(x) = \exp{(-|x|)} $$
which is proportional to a Lorentzian function:
$$\hat{f}(w) = \frac{1}{1+w^2} $$
(I am ignoring constants, etc., which are not important to this discussion.)  Note that $\hat{f}(w) \approx 1/w^2 (w \rightarrow \infty)$.  The algebraic, rather than exponential, behavior at $\infty$ is characteristic of a type of discontinuity.  In this case, there is a discontinuity in the derivative.  For a straight discontinuity, there is a $1/w$ behavior at $\infty$.  (Note the step function and its transform which is proportional to $1/w$ at $\infty$.)  For a discontinuity in the 2nd derivative, there is a $1/w^3$ behavior at $\infty$.  And a discontinutiy in the $k$th derivative of $f(x)$ translates into a $1/w^{k+1}$ behavior of the Fourier transform at $\infty$.
No, I do not have a proof of this, so I am talking off the cuff from my experiences some moons ago.  But I am sure this is correct for the functions we see in physics.
Also note that I define the Fourier Transform here as 
$$\hat{f}(w) = \int_{-\infty}^{\infty} dx \: f(x) \exp{(-i w x)} $$
