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Thread: continuity

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by gokole View Post
    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|.
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  3. #3
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    thankx for ur help I really appreciate it..
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