I am quite new in analysis, and just have two questions in continuous differentiability. 1, If log(f) is continuously differentiable, can I say that f is also when f is strictly positive? 2, I know that using implicit function theorem, we can show the continuous differentiability by the Jacobian matrix. Is there any sufficient conditions that I can use to show that? As the implicit function theorem failed in my experiment, I'd like to apply other theorems. Thanks.
The composition of $C^1$ functions is $C^1$, in the sense that:
If $\Omega_1 \subset \Bbb R^n$ and $\Omega_2 \subset \Bbb R^m$ are two open sets, and if $f: \Omega_1 \to \Bbb R^m$, $g: \Omega_2 \to \Bbb R^d$ are two maps such that $f \in C^1(\Omega_1)$ and $g \in C^1(f(\Omega_1))$, then the map $g \circ f$ is of class $C^1$ on $\Omega_1$.
Assuming that $f: \Omega \subset \Bbb R^n \to \Bbb R$ (where $\Omega$ is an open subset of $\Bbb R^n$), if $\log \circ f \in C^1(\Omega)$, then, as $\exp \in C^1(\Bbb R)$, we get $f = \exp \circ (\log \circ f) \in C^1(\Omega)$.