can there be a function which is discontinuous at a point but differentiable at that point? explain giving examples?

$$\begin{gathered} function given by, \\ f\left(x\right)=\left\{\begin{array}{l}{x}^2 , if x is rational\\ -x^2 , if x is irrational\end{array}\right. \end{gathered}$$
Now this function is not continuous because when we will draw  graph for rational value and irrational value there will be break point for every point because we are taking two different value one is rational and other is irrational.

$$\begin{gathered} Now finding \\ \underset{h\to 0}{\lim } \frac{f(0+h)-f\left(0\right)}{h} \\ Now the diffrential quotient can be written as, \\ \frac{f\left(h\right)-f\left(0\right)}{h}=\left\{\begin{array}{l}h , if h is rational\\ -h , if h is irrational\end{array}\right. \\ Here f\text{'}\left(0\right) exists and is 0. \end{gathered}$$