# Local property of smooth functions with compact support

I was wondering if the following is true and if this result can be strengthened to more general "growth rates" than logarithms. Let $$\phi \in C_{0}^{\infty}(\mathbb{R}) = \{ f \in C^{\infty}(\mathbb{R}) |$$ $$supp(f)$$ is compact$$\}$$. Then $$\lim_{\epsilon \downarrow 0} \left( \phi(\epsilon) - \phi(-\epsilon) \right)ln(\epsilon) = 0$$. Below is what I believe to be a proof.

Since $$\phi$$ is smooth, let $$K > 0$$ be a Lipschitz constant for $$\phi$$ on $$[-1,1]$$. Then for $$\epsilon < 1$$, we have \begin{align*} |\phi(\epsilon) - \phi(-\epsilon)||ln(\epsilon)| &\leq 2 K \epsilon ln(\epsilon) \to 0 \end{align*} as $$\epsilon \downarrow 0$$.

In general, this should work for any function that goes as $$\epsilon ^ p$$ where $$p > -1$$ correct? Then $$p = 1$$ has the counterexample $$\phi \equiv \epsilon$$ in a neighborhood of $$0$$. Also, would this rate also be generalizable?

• $\phi(\epsilon)-\phi(-\epsilon) = \int_{-\epsilon}^\epsilon \phi'(x)dx$ so $|(\phi(\epsilon)-\phi(-\epsilon))\ln(\epsilon)| \le 2 |\epsilon\ln(\epsilon)| \sup_{x \in [-1,1]} |\phi'(x)|$ – reuns Dec 8 '18 at 21:28
• Yeah that would be essentially the same proof right? The Lipschitz constant for a differentiable function is in general $sup_{x \in [-1,1]} |\phi'(x)|$ right? By the mean-value theorem – TylerMasthay Dec 9 '18 at 21:35