f is defined everywhere, there can't be any values of x where f is discontinuous because it is undefined. That means we need to look for places where the limit of f is undefined or disagrees with the value of f. The only place this could possibly happen is x = 0. We will investigate. The left-hand limit is
The right-hand limit is
The one-sided limits agree,
Alas, this disagrees with the function value f(0) = 0. The function is discontinuous at x = 0.