Absolute value: d/dx(|x|) = x/|x| = sgn(x) for x != 0. Not differentiable at x = 0.

More generally: d/dx(|f(x)|) = f'(x) * f(x)/|f(x)| = f'(x) * sgn(f(x)) for f(x) != 0.

Greatest integer function: d/dx([x]) = 0 for all non-integer x. Not differentiable at integers.

Fractional part: d/dx({x}) = 1 for all non-integer x. Not differentiable at integers.

Signum function: d/dx(sgn(x)) = 0 for x != 0. Not differentiable at x = 0.

Max and Min functions: max(f,g) = $\frac{f+g}{2}$ + |f-g|/2 min(f,g) = $\frac{f+g}{2}$ - |f-g|/2 Differentiable everywhere except where f(x) = g(x) (corner points).