0
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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 $f = \exp \circ (\log \circ f) \in C^1(\Omega)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.