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. doesn't give everything I'm looking for, because I want to show differentiability everywhere.
Best Answer
I'm not super sure if I understand your question completely, because the continuity of partial derivatives can be extended to the entire space (and not just a single point). Perhaps it will be helpful to look at the actual statement of the theorem, which we discuss below.
One can use the following theorem from Munkres' Analysis on Manifolds that is sufficient for differentiation but not necessary:
Theorem 6.2. Let $A$ be open in $\mathbb{R}^m$. Suppose that the partial derivatives $D_j f_i (x)$ of the component functions of $f$ exist at each point $x$ of $A$ and are continuous on $A$. Then $f$ is differentiable at each point of $A$.
In your case, take $\frac{d}{dx}, \frac{d}{dy}$ and see if they are continuous. If so, then your function $f$ is differentiable at each point.