Smoothness of functions global and along curves over ultra-metric fields
The article is devoted to the investigation of smoothness of functions $f(x_1,...,x_m)$ of variables $x_1,...,x_m$ in infinite fields with non-trivial multiplicative ultra-norms, where $m\ge 2$. Theorems about classes of smoothness $C^n$ or $C^n_b$ of functions with continuous or bounded uniformly continuous on bounded domains partial difference quotients up to the order $n$ are investigated. It is proved, that from $f\circ u\in C^n({\bf K},{\bf K}^l)$ or $f\circ u\in C^n_b({\bf K},{\bf K}^l)$ for each $C^{\infty}$ or $C^{\infty }_b$ curve $u: {\bf K}\to {\bf K}^m$ it follows, that $f\in C^n({\bf K}^m,{\bf K}^l)$ or $f\in C^n_b({\bf K}^m,{\bf K}^l)$ respectively. Moreover, classes of smoothness $C^{n,r}$ and $C^{n,r}_b$ and more general in the sense of Lipschitz for partial difference quotients are considered and theorems for them are proved.