Similar to this problem on another forum.

Show f(x) = { x/2 if x rational , x if x irrational is not differentiable at 0

Basic idea is that there are two possible (secant) slopes: 0 and 1, and you can always find values of x satisfying both in any (-delta, delta) \ {0}, delta > 0.

I don't entirely follow your potential proof. Shouldn't you be subtracting L instead of 0 inside the first absolute value?