Making discontinues function, continues.

[burak100]

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.

 

[jbunniii]

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?).