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Making discontinues function, continues.

  1. Oct 14, 2013 #1
    making discontinues function, continues.!!

    1. The problem statement, all variables and given/known data

    Given a function g, which is not continuous everywhere and g is increasing. The problem is how to approach to this function to make it continuous.

    2. Relevant equations



    3. The attempt at a solution

    I am not sure but one way maybe using average value but the problem is; to use the average value, g should be continuous.
     
  2. jcsd
  3. Oct 14, 2013 #2

    jbunniii

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    Says who? We can certainly form a function ##h## by averaging ##g## over a small neighborhood of each point:
    $$h(x) = \int_{x - \epsilon}^{x+\epsilon} g(y) dy$$
    Since ##g## is increasing, it can only have jump discontinuities, and at most countably many of them (proof?). Therefore, the Riemann integral is well defined at every ##x##. It's straightforward to show that ##h## is continuous everywhere (proof?).
     
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