Let $\mathbb H^n=\{x\in\mathbb R^n:x_n\geq 0\}$. I saw the following definition:
- A function $f:U\to\mathbb R$ defined on an open subset of $\mathbb H^n$ is smooth iff it has a smooth extension to an open subset of $\mathbb R^n$.
However in the halfspace $\mathbb H^n$ the notion of partial derivatives still makes sense so for example even if for $y\in U$ we have $y_n=0$ we could define $$\frac{\partial f}{\partial x_n}(y)=\lim_{h\to0,\,h\geq 0}\frac{f(y+h\cdot e_n)-f(y)}{h}$$
if the right hand side exists. Clearly if $f$ is smooth as in 1. all partial derivatives exist.
The question is: If all partial derivatives of all order exist, is $f$ smooth as in 1.?