Log-Lipschitz of a density function

Question

The Log-Lipschitz continuous condition implies that for a given function $f$ exists a non-negative constant $L\leq \infty$ such that

$$ |\log f(x) - \log f(y)| \leq L |x-y| \quad \quad \forall x,y\in \mathbb{R}. $$

I noticed that this condition is not satisfied by the Gaussian function. I am wondering if it exists any density function (therefore a non-negative function such that $\int_{\mathbb{R}} f(x)\mathrm{d} x = 1$) which satisfies the Log-Lipschitz condition. I would be glad to see one example of such density.

Thanks.

Answer 1

Exponential density has this property on $(0,\infty)$ and $\frac 1 2 e^{-|x|}$ has this property on the whole line.