Let $f$ be a $C^{m}(\mathbb{R}^{n})$ function, that is a function which is $m$ times continuously differentiable. Is it true that $$ |\nabla f(x)| \leq \frac{C}{|x|^{n}}$$ for each $x \in \mathbb{R}^{n}$ and $C>0$?
I think this is true due to the fact that each component of $f$ is integrable because of being the derivative of $f$ and $g(x)=1/|x|^{n}$ isn't integrable in $\mathbb{R}^{n}$ but I can't write this in a rigurous way.
Edit: I can take extra assumptions: $f(-x)=-f(x)$ and $f(ax)=a^{-n+1}f(x)$.
Thank you.