In this question, the expression of f(x) where x is not 0 has a limit of 0 when x approaches 0, so it is continuous.

On the derivative at 0: Note that we can't take the derivative first and then approach 0 with the above formula, but define it with the derivative, lim (f(x)-0)/(x-0), which is found to be divergent, so there is no derivative at 0.

Since there is no point 0 in the derivative function, it cannot be continuous.

Please refer to the picture for details. Please forgive me for adding some marks to prevent stealing pictures.