In this paper, we are interested in testing whether the volatility process is constant or not during a given time span by using high-frequency data with the presence of jumps and market microstructure noise. Based on estimators of integrated volatility and spot volatility, we propose a nonparametric procedure to depict the discrepancy between local variation and global variation. We show that our proposed test statistic converges to a standard normal distribution if the volatility is constant, and diverges to infinity otherwise. Simulation studies verify the theoretical results and show a good finite sample performance of the test procedure. We also apply our test procedure to some real high-frequency financial datasets. We observe that in almost half of the days tested, the assumption of constant volatility within a day is violated. And this is due to that the stock prices in the periods near the opening and closing are highly volatile and account for a relatively large proportion of intraday variation.