Proving Continuity: Discontinuity Math Help and Tips

[Bleys]

Homework Statement

f is a function with the property that every point of discontinuity is removable. There are infinitely many such points in f's domain. Define $$g(x) = \lim_{ y \to x } f(y)$$. Prove g is continuous

The Attempt at a Solution

I wanted to maybe conclude something from showing g is bounded but I didn't really get anything there. I was wondering if you could give me a hint, but DON'T GIVE ME A SOLUTION yet. Should I be using straight forward definition or take some other route?

 

[ystael]

You should be able to do this by following the definitions in a straightforward manner, more or less using only the concepts of limit and continuity. The key is that when you are examining the behavior of $$g$$ on an interval, you can transfer statements about $$g$$ at a point to statements about $$f$$ on a smaller interval.