The following system is an example in a book of dynamical system(in the section about Hopf Bifurcation).
$ \begin{align} \dot{x}=\mu x- y-x\sqrt{x^2+y^2} \\ \dot{y}=x + \mu y-y\sqrt{x^2+y^2} \end{align} $
But I don't understand why it can be called a $C^1$-system, i.e., the vector field $f:R^2\times R\to R^2$ defined by this systemm belongs to $C^1(R^2\times R)$, i.e., all of the first partial derivatives with respect to $x,y$ and $\mu$ are continuous for all $x,y$ and $\mu$.
I set it as an exercise, and found that $ \frac{\partial f}{\partial x}= \left( \begin{array}{c} \mu-(\sqrt{x^2+y^2}+\frac{x^2}{\sqrt{x^2+y^2}}) \\ 1-\frac{xy}{\sqrt{x^2+y^2}} \\ \end{array} \right) $ This is not even defined at $(0,0,\mu)$. Then why $f\in C^1(R^2\times R)$?
Edit: The original question finally boils down to another question:
What exactly is the definition of $C^1$ functions, or more generally $C^k$ functions? Are the $k$-th derivatives allowed to have removable-discontinuity points?