Suppose we have a function $g: \mathbb R^2 \to \mathbb R$ and $$\nabla g(u,v)=(5v^4-2u\exp(v-u^2), \exp(v-u^2)+20uv^3), (u,v)\in\mathbb R^2$$
Can the function $g$ be twice differentiable, i.e. does the second-order total differential exist?
One can see immediately, that the first order total differential, the gradient function $\nabla g$, is partially differentiable both in terms of $u$ and $v$, but we are trying to find total derivatives.
Is there any other way than trying to differentiate in therms of $u(v)$ and $v(u)$? Is this correct approach at all?
It has something to do with continuous and non-continuous, can I simply state that since both components of the gradient are continuous, the second order total derivative exist?
