In this paper, we consider the problem of testing for a change of the parameter function ${ heta}(t)$ that may have a discontinuity at some unknown point ${ au}$. We introduce a varying-h moving estimate to test the null hypothesis that ${ heta}(t)$ is continuous against the alternative that ${ heta}({ au}-){ eq}{ heta}({ au}+)$. Simulation results are provided for illustration.