Theorem: Suppose that $$f(z)=u(x,y)+i\,v(x,y)$$ and that $f’(z)$ exists at a point $z_0=x_0+i\,y_0$. Then the first order partial derivatives of $u$ and $v$ must exists at $(x_0,y_0)$, and they must satisfy the Cauchy-Riemann equations $$u_x=v_y,\quad u_y=-v_x$$ there. Also, $f’(z_0)$ can be written $$f’(z_0)=u_x+i\,v_x,$$ where these partial derivatives are to be evaluated at $(x_0,y_0)$.
I understand that these equations provide necessary conditions for the existence of $f’(z_0)$ which is weaker than sufficient conditions. So my book, in the examples, states that, if the Cauchy Riemann equations are not satisfied then the derivative a function does not exist. Therefore, my question is, would I be able to use this theorem to find points in which a function is not differentiable?