Let's consider two cases:
For $f:\mathbb{R}^2 \to \mathbb{R}$, where the domain may be some open subset in $\mathbb{R}^2$, define its sections to be functions in $\{ f(,x_2), f(x_1,), \forall x_1,x_2 \in \mathbb{R}\}$.
Do differentiablility of $f$ and differentiability of each of its sections in imply each other, or just one-directional implication, or neither?
For $f:\mathbb{R} \to \mathbb{R}^2$, where the domain may be some open subset in $\mathbb{R}$, define its components to be functions $\{f_i:\mathbb{R} \to \mathbb{R}, i=1,2, \text{ s.t. }f(x)=[f_1(x), f_2(x)], \forall x \in \mathbb{R} \}$.
Do differentiablility of $f$ and differentiability of each of its components imply each other, or just one-directional implication, or neither?
- Are answers to part 1 and part 2 the same when $f:\mathbb{R}^n \to \mathbb{R}$ and $f:\mathbb{R} \to \mathbb{R}^n$?
Thanks and regards!
PS: In part 1, I was hoping to ask for a more general case $f:\mathbb{R}^n \to \mathbb{R}$, but didn't know how to write down a single formula for defining all its sections. Its sections are defined by fixing all except one of the n components of the domain variable and varying the exceptional one. Do you have some idea?