If a function has, say, partial derivatives up to order n, can you conclude continuity of some or all derivatives of lower order?
Especially, if a function has partial derivatives of any order is it automatically smooth?
If a function has, say, partial derivatives up to order n, can you conclude continuity of some or all derivatives of lower order?
Especially, if a function has partial derivatives of any order is it automatically smooth?
Consider the function $f:\mathbb{R}^2\to \mathbb{R}$ defined by $f(x,y):=\frac{xy}{x^2+y^2}$ for $(x,y)\ne (0,0)$ and $f(0,0):=0$.
This function has partial derivatives of any order anywhere in $\mathbb{R}^2$ (note that on the $x$- and $y$- axes it is just 0), but it is not even continuous.
Edit: I see now that this is not a counterexample because $\frac{\partial f}{\partial x}$ is no longer zero on the $y$-axis, so $\frac{\partial^2 f}{\partial x\partial y}$ does not exist. So here is a corrected counterexample: $f(x,y):=e^{-\frac{(x^2+y^2)^2}{x^2 y^2}}$ for $x\ne 0, y\ne 0$ and $f(x,y)=0$ on the coordinate axes. This function is not continuous at 0 (consider the restriction to the line $x=y$), but it is smooth outside of 0, and all derivatives still have the property that they are 0 on the coordinate axes. Hence all partial derivatives of any order also exist in 0.