Differentiability of $\mathrm{max}(x, y)$ at $x=y$ I am trying to figure out differentiability of $\mathrm{max}(x, y)$. Intuitively, it should not be differentiable at $x=y$, since it changes direction "non-smoothly" at those points. 

I can not, however, find a way to prove this algebraically.
Hint: Write

$$\operatorname{max}(x,y) = \frac{x + y + |x-y|}{2}$$
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$$
\partiald{\max\pars{x,y}}{x} = \half\bracks{1 + \sgn\pars{x - y}}
$$