Continuity of Integral with Fixed Variables in Lebesgue Integration

[Mystic998]

Homework Statement

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?

Homework Equations

N/A

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.

 

[StatusX]

But that epsilon depends on y.