# Continuity of an integral

1. Apr 4, 2007

### Mystic998

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

For reference, this is chapter 11, problem 12 of Rudin's Principals of Mathematical Analysis.

Suppose $$|f(x,y)| \leq 1$$ if $$0 \leq x \leq 1, 0 \leq y \leq 1$$; for fixed x, f(x,y) is a continuous function of y; for fixed y, f(x,y) is a continuous function of x.

Put $$g(x) = \int_{0}^1 {f(x,y) dy}, 0 \leq x \leq 1$$.

Is g continuous?

2. Relevant equations

N/A

3. The attempt at a solution

To me it seems like this is obviously continuous since within an integral you're essentially working with a fixed y, so you can find some delta such that |f(x,y) - f(a,y)| is less than any positive epsilon, then the inequality you actually need follows easily. But it just seems way too easy.

Anyway, hope I didn't mangle the TeX. I'm not used to using it.

Just as a note, I'm referring to Lebesgue integration.

Last edited: Apr 4, 2007
2. Apr 4, 2007

### StatusX

But that epsilon depends on y.