# Integrability implies continuity at a point

1. Jul 3, 2012

### AlwaysCurious

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

If f is integrable on [a,b], prove that there exists an infinite number of points in [a,b] such that f is continuous at those points.

2. Relevant equations

I'm using Spivak's Calculus. There are two criteria for integrability that could be used in this proof (obviously, they have been shown to be equivalent). The first is the usual inf(upper sums) = sup(lower sums) one, and the second is that for every epsilon greater than zero, there is a partition P such that the upper sum over P minus the lower sum over P is less than epsilon.

3. The attempt at a solution

I haven't made much progress - obviously the second definition seems a bit easier to use, and I have figured out that if you prove that if it is continuous at one point in the interval, it is continuous at an infinite number of points on the interval. So the problem is reduced somewhat.

Any hints?

Thank you!

2. Jul 3, 2012

### micromass

Staff Emeritus
This is part (e) of the exercise in Spivak. Did you already show (a)-(d)?? Where are you stuck exactly??

3. Jul 3, 2012

### AlwaysCurious

Yeah, I didn't want to do those parts/look at them because I knew that he was spelling out the solution - so I attempted to all do it in my head, away from the book paper but wasn't able to get anywhere.

I eventually just looked at the book's solution (parts a through d).

As a side note, what do you think of Hubbard's Vector Calculus, Linear Algebra, and Differentiable Forms? You recommended it a while ago. What are its strengths, what are its weaknesses? How does it compare to other good books on similar subjects?

Thank you so much!