Summary: We combine the wavelet method and the limit-theorem to test the presence of jumping-points in a nonparametric function which is observed with unit-root noise. First, the limit distribution of the wavelet coefficients of the noise is derived using the limit-theorem, and then, the statistic of detection is determined. When the null hypothesis holds, we obtain the critical values at any scale, prove the consistency of wavelet detection and give the threshold of wavelet coefficients. When the alternative hypothesis holds, the consistent estimation of the numbers and locations of jumping points are given and the rate of convergence is obtained. Simulation study and real data analysis support our method. Finally, we compare our method with the “UNI”-method and the “GOF”-method. Asymptotic results show that our method is more powerful in detecting the jumping points of a nonparametric function with unit-root noise.