A useful result related with zeros of continuous compactly supported mother wavelets

In the last two decades, wavelet bases and associated methodologies have become quite important in many domains, such as signal and image processing, harmonic analysis, statistics, and so on. More recently, they also turn out to be quite useful in the probabilistic framework of stochastic processes, in which, among other things, they allow to obtain fine results concerning erratic sample paths behavior.
The goal of our article is to derive a result, related with zeros of continuous compactly supported mother wavelets, which is useful in this probabilistic framework. More precisely, let $\psi$ be any arbitrary such wavelet; we show that being given an arbitrary point $x_0\in\R$ there always exists at least one integer $k_{x_0}\in\Z$ such that $\psi(x_0-k_{x_0}) \neq 0$.