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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!
