Integration Real Analysis Problem

Question

Ive got an interesting question:

Let $f$ be continuous on [−1, 1] \ {0}. Does it follow that $f$ is integrable on [−1, 1]?

Suppose that −1 < a < b < 0. Is f integrable on [a, b]?

I am not sure how to approach this question so any help towards the answer will be appreciated.

Answer 1

Hint: for the first one, take $f(x) = \frac 1x$ What happens? For the second one, what do you know about integrability of continuos functions on closed intervals?

Answer 2

I think the most appropriate answer is indicated for you the book "Principles of Mathematical Analysis- Rudin". What you want to know is clearly explained in chapter 6. Another option, also very good and perhaps more detailed, you find in "Introduction to real analysis - Bartle"