# Making discontinues function, continues.

1. Oct 14, 2013

### burak100

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. Oct 14, 2013

### jbunniii

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?).