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.
 A: Hint: Write
$$\operatorname{max}(x,y) = \frac{x + y + |x-y|}{2}$$
A: Take any point $(a,a)$ and prove that the partial derivative does not exist. The partial derivative is $\lim_{x\rightarrow a} \frac{f(x,a)-f(a,a)}{x-a}$. The left limit for this is
$$\lim_{x\uparrow a}\frac{\max\{x,a\}-a}{x-a}=\lim_{x\uparrow a}\frac{a-a}{x-a}=0$$
while the right limit is
$$\lim_{x\downarrow a}\frac{\max\{x,a\}-a}{x-a} = \lim_{x\downarrow a}\frac{x-a}{x-a}=1$$
A: Let $f(x,y)=\max \{x,y\}$. Now look at $f$ on the section $y=-x$:
$$
f(x,-x)=\max \{x,-x\}=|x|.
$$
This function is not differentiable as a function of the variable $x$, so $f$ can't be differentiable.
A: $\newcommand{\+}{^{\dagger}}%
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$$
\partiald{\max\pars{x,y}}{x} = \half\bracks{1 + \sgn\pars{x - y}}
$$
