Help Solve Integration Theory Assignment

[mooberrymarz]

Guyz please help me !

hey! Could any of u please help me with this question. Its for my general integration theory assigment.

'Use an argumetn by contradiction to establish the following claim:
If f is continuous on R and $$\int |f|= 0,then f = 0.[\tex]<br /> <br /> thanx$$

 

[mathman]

Rough argument. If f is not 0 at some point, it must be not 0 in an interval around that point (continuity). Therefore the integral of |f| will be >0.

 

[M98Ranger]

!

Sure, I'd be happy to help! To begin, let's first define what it means for a function to be continuous on R. A function f is continuous on R if it is continuous at every point in the set of real numbers, R. This means that the limit of f(x) as x approaches any real number a, is equal to f(a).

Now, let's assume that the claim is false. This means that there exists a function f that is continuous on R and has an integral of 0, but f is not equal to 0. In other words, there exists at least one point a in R where f(a) is not equal to 0.

Since f is continuous on R, this means that it is also continuous at a. Therefore, we can use the definition of continuity to say that the limit of f(x) as x approaches a is equal to f(a).

Now, since f(a) is not equal to 0, this means that there exists a small interval around a where f(x) is also not equal to 0. Let's call this interval (a - δ, a + δ), where δ is a positive number.

Next, we can use the definition of the integral to say that the integral of f over this interval is equal to the limit of the Riemann sum as the partition size approaches 0. Since f is not equal to 0 on this interval, this means that the Riemann sum will also not approach 0.

Therefore, we can conclude that the integral of f over this interval is not equal to 0. But this contradicts our initial assumption that the integral of f over R is equal to 0.

Thus, our assumption that f is not equal to 0 must be false. In other words, f must be equal to 0 for the claim to hold true.

I hope this helps! Let me know if you have any other questions. Good luck with your assignment!