Continuity of Integral with Fixed Variables in Lebesgue Integration

In summary, the problem is asking if the function g(x) defined as the integral of f(x,y) over the interval [0,1] is continuous. The given conditions suggest that f(x,y) is a continuous function of both x and y, leading to the conclusion that g(x) is also continuous. The use of Lebesgue integration further supports this conclusion.
  • #1
Mystic998
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Homework Statement



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

Suppose [tex] |f(x,y)| \leq 1 [/tex] if [tex] 0 \leq x \leq 1, 0 \leq y \leq 1 [/tex]; 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 [tex] g(x) = \int_{0}^1 {f(x,y) dy}, 0 \leq x \leq 1 [/tex].

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.
 
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  • #2
But that epsilon depends on y.
 

1. What does continuity of an integral mean?

Continuity of an integral refers to the property of a function where small changes in the input variable result in small changes in the output variable. In other words, a continuous integral is a function that can be drawn without lifting the pen from the paper.

2. How is continuity of an integral mathematically defined?

Mathematically, continuity of an integral is defined as a function f(x) being continuous at a point x=a if the limit of f(x) as x approaches a is equal to the value of f(a). This can be written as lim_{x \to a} f(x) = f(a).

3. What is the importance of continuity of an integral?

The continuity of an integral is important because it allows for the use of various mathematical techniques, such as the fundamental theorem of calculus, to evaluate the integral. It also ensures that the function is well-behaved and avoids any abrupt changes or discontinuities.

4. How is continuity of an integral related to differentiability?

The continuity of an integral is a necessary condition for differentiability. A function must be continuous in order to be differentiable at a point, but it is possible for a function to be continuous without being differentiable. Therefore, continuity is a weaker condition than differentiability.

5. Can a function with a discontinuous integral still be integrable?

Yes, a function with a discontinuous integral can still be integrable. This is because the integral is defined as the limit of a sum of infinitesimal rectangles, and these rectangles can still have finite areas even if the function has a discontinuity. However, the function must still satisfy other conditions, such as boundedness, in order for the integral to exist.

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