rational function of non vanishing Hölder continuous functions

Question

Conside Hölder continuous functions $f_1,…,f_m:\mathbb{R}^n\rightarrow \mathbb{R}$ (with Hölder coefficient $\alpha$). The claum is now that any rational function of $f_1,…,f_m$ with non vanishing denominator is also Hölder continuous (with coefficient $\alpha$). The author says this is trivial, but I do not see why this should be trivial!? (I could not even prove the easiest case $f(x)/g(x)$)

Do you have an idea by which it is easily proven?

Best regards.

Answer 1

Such as stated, the result is not true. Take for instance $f_1(x)=e^{-x^2}$, $f_2(x)=e^{-2x^2}$, which are Lipschitz. Then $f_1(x)/f_2(x)=e^{x^2}$, which is not (globally) Lipschitz. Two conditions that will make the result true:

1. The $f_i$ are bounded
2. The denominator is bounded away from $0$

Answer 2

I it is not true in general. For example $f_1(x)= exp( \sqrt{x})$ and $f_2(x)= 5 \,exp( \sqrt{x})$. The ratio is 5, so the Holder exponent is not conserved.