I came across such problem:
$g:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^1$-function with $g(\theta+\pi)=-g(\theta)$ for all $x$. Define a function $f: \mathbb{R}^2\rightarrow \mathbb{R}$ as $$f(x,y)=f(r\cos{\theta},r\sin{\theta})=r\cdot g(\theta). $$
Show that $f$ is differentiable everywhere in $\mathbb{R}^2$ except possibly at $(x,y)=(0,0)$ and also that all the directional derivatives of $f$ exist at $(0,0)$
Can anyone show me the differentiability here just by the definition that
$\displaystyle \lim_{||h||\rightarrow 0}{ \frac{||f(x_0+h)-f(x_0)-J\cdot h||}{||h||}}=0$
Since there is change of variable involved as well and I don't know what the use of the periodicity of $g$ here, I am confused even more about the question.
I'll really appreciate if someone can show me the computation process, because I really need a concrete example to understand this well. Thanks :)