I'm trying to get an intuition for smooth manifolds, and in particular the smoothness of transition functions. I haven't done that much calculus on $\mathbb{R}^n$ before, and would like to practice proving that some functions $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ are smooth or not smooth. I believe this is just a case of checking all partial derivatives $\frac{\partial f_i}{\partial x_j}$ exist to arbitrary order. Is this correct?
To facilitate practising this, does anyone know of a good selection of examples I could try? All the textbooks on differential geometry seem to assume I'm already completely comfortable with this, so provide no such exercises! Could someone point me at a good resource online or a good book? Alternatively could someone provide some interesting problems/examples in an answer?
Many thanks in advance.