We can differentiate a function with a modulus operator.
y = absolute (x) = | x |.
The function is continuous at every point and is differentiable too at every point except at x = 0.
Some functions with a modulus operator may not be differentiable at some points.
y = | tan x |
ie., y = tan x 0 <= x < = Pi/2
= - tan x - Pi/2 <= x <= 0 or ,
or tan z, z = -x, 0 <= z <= Pi/2
at all points except at y= x = 0, this function is continuous and differentiable.
at x = 0 =y,
dy/dx on x = 0+, is sec^2 x = sec^2 0 = 1
dy/dx on x = 0- or z = 0+ , is sec^2 z = 1
Hence, the function is differentiable at x = 0.