How to find the non-differentiable point(s) of a given continuous function?

For example, the non-differentiable point of the function $f(x)=|x|$ is at $x=0$.
How to find the non-differentiable points of a continuous function that is defined numerically?

Answer

If we define f[x] e.g. like this:

f[x_] := Abs[x]

the following returns interesting points:

Reduce[ 
    Limit[(f[x + h] - f[x])/h, h -> 0, Assumptions -> x ∈ Reals, Direction -> -1] != 
    Limit[(f[x + h] - f[x])/h, h -> 0, Assumptions -> x ∈ Reals, Direction -> 1],  x]

x == 0

Let's try another function defined with Piecewise, e.g.

g[x_] := Piecewise[{{x^2, x < 0}, {0, x == 0}, {x, 1 > x > 0}, 
                    {1, 2 >= x >= 1}, {Cos[x - 2] + x - 2, x > 2}}]

then we needn't use Assumptions in Limit:

Reduce[ Limit[ (g[x + h] - g[x])/h, h -> 0, Direction -> -1] != 
        Limit[ (g[x + h] - g[x])/h, h -> 0, Direction -> 1], x]

 x == 0 || x == 1 || x == 2

pts = {x, g[x]} /. {ToRules[%]};

Plot[ g[x], {x, -5/4, 3}, PlotStyle -> Thick, 
      Epilog -> {Red, PointSize[0.023], Point[pts]}]

One should be careful when working with Piecewise since Reduce may produce errors when weak inequalities (LessEqual) are involved. For this reason we added {0, x == 0} in the definition of the function g.