How to show that a test statistics can't reject H0 with a small change in the hypothesized value

Question

In the homework, suppose the null hypothesis is given by $H_{0\text{ }}: \alpha =\alpha _{0}$, and a test statistics $T\left( x\right) $ is proposed. The question is to show that $T\left( x\right) $ can't reject $H_{0\text{ }}$ if $\alpha =\alpha _{0}+\varepsilon $ where $\varepsilon $ is a small number, I am a little lost at this. Any help will be highly appreciated! Thanks!