I'm given a function like $(x^2+y^2)sin(1/\sqrt{x^2+y^2})$ that's $0$ at $(x,y)=0$, and the function is obviously not of class $C^1$, is there any way to prove that it's differentiable other than the limit method showing $\exists B,lim\_{h\rightarrow 0}(f(a+h)-f(a)-Bh)/||h||=0$ ?

Otherwise, is there any particular (general) tricks to proving functions like this are differentiable through the limit method?

Note: [Proving a scalar function is differentiable at the origin but that its partial derivatives are not continuous at that point.](https://math.stackexchange.com/questions/607104/proving-a-scalar-function-is-differentiable-at-the-origin-but-that-its-partial-d) doesn't give everything I'm looking for, because I want to show differentiability everywhere.