Please could you help me guys. My professor gave me the following question and I don't know how to demonstrate the $\displaystyle \epsilon$ and $\displaystyle \delta_\epsilon$ in this question.

Question. if you have $\displaystyle f$ is a measurable real-valued function on the unit circle $\displaystyle \mathbb{T}$ prove that the following function is right continuous

$\displaystyle m_{f}(x)=|\{ t: f(t) \leq x \}|,$ where $\displaystyle -\infty < x < \infty$