# continuity

• July 23rd 2009, 11:55 AM
gokole
continuity
Please could you help me guys. My professor gave me the following question and I don't know how to demonstrate the $\epsilon$ and $\delta_\epsilon$ in this question.

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

$m_{f}(x)=|\{ t: f(t) \leq x \}|,$ where $-\infty < x < \infty$
• July 23rd 2009, 01:02 PM
Opalg
Quote:

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 $\epsilon$ and $\delta_\epsilon$ in this question.

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

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

For $x\in\mathbb{R}$, let $S_x = \{t\in\mathbb{T}:f(t)\leqslant x\}$. Then $S_a = \textstyle\bigcap_{x>a}S_x$ (because $t\leqslant x$ for all $x>a$ implies $x\leqslant a$). Since $\mathbb{T}$ has finite measure, it follows that $|S_a| = \lim_{x\searrow a}|S_x|$.
• July 23rd 2009, 04:02 PM
gokole
thankx for ur help I really appreciate it..