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Making discontinues function, continues.

  • Thread starter burak100
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  • #1
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making discontinues function, continues.!!

Homework Statement



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.

Homework Equations





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.
 

Answers and Replies

  • #2
jbunniii
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to use the average value, g should be continuous.
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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