1. ## continuity

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 $\displaystyle -\infty < x < \infty$

2. Originally Posted by gokole
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 $\displaystyle -\infty < x < \infty$
For $\displaystyle x\in\mathbb{R}$, let $\displaystyle S_x = \{t\in\mathbb{T}:f(t)\leqslant x\}$. Then $\displaystyle S_a = \textstyle\bigcap_{x>a}S_x$ (because $\displaystyle t\leqslant x$ for all $\displaystyle x>a$ implies $\displaystyle x\leqslant a$). Since $\displaystyle \mathbb{T}$ has finite measure, it follows that $\displaystyle |S_a| = \lim_{x\searrow a}|S_x|$.

3. thankx for ur help I really appreciate it..