Suppose the function $f : \Bbb R \to \Bbb R$ is continuously differentiable and define another function $g$ as $$g(x,y) := f \left(\sqrt{x^{2} + y^{2}}\right)$$ Under what condition is $g$ differentiable at $(0,0)$?
I am thinking of this one in the lines of Jacobi map of a composition of two functions, where the second function is the square root function. But, it doesn't get me anywhere because the Jacobi matrix of the square root function doesn't exist at (0,0).
Related Question: Given any function between any two finite-dimensional Euclidean spaces, does the existence of the Jacobi matrix at a point guarantee the differentiability of the function at that point?