The continuity of a function, at a certain point, happens when the left hand limit of the function is equal to the right hand limit of the function at that point. These limits must be equal to the value of function at that point.

Notice that the questionable point is x = 0.

`lim_(x-gt0)x/|x| = lim_(x-gt0) (xlt0)1/|-1| = -1`  (use l'Hospital's theorem)

`lim_(x-gt0)x/|x| = lim_(x-gt0) (xgt0)1/|1| = 1`

Since the left hand limit `!=`  right hand limit, the function is discontinuous at x = 0.