Let $g(x,y)$ be a continuous odd function on the unit circle. We define a function on plane as a linear interpolation of $g(x,y)$ on each straight line passing through $0$: $f(x,y)=|x,y|*g(\frac{(x,y)}{|x,y|})$ if $(x,y) \ne 0$ and $f(x,y)=0$ if $(x,y) = 0$.
Statement: $f$ is not differentiable at zero, except for the case $g ≡ 0$. Construct a disproving example to this statement. Give the correct wording and prove it.
Let $g(x,y) = x$, then $f(x,y) = x$. Clearly, $g$ is odd and $f$ is differentiable.