Is f Integrable? Prove It or Give an Argument

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In summary, the function f is defined on [0,1] so that f(1/n)=n^(-1/2) for n=1,2,3,... and f(x)=0 if x is not a reciprocal of a positive integer. The question is whether f is integrable on [0,1] and whether it is continuous. After analyzing the properties of f, it can be concluded that f is integrable, both Riemann and Lebesgue. To show Riemann integrability, it can be divided into m subintervals where most of them will not contain a point where x=1/n, and the upper and lower Darboux sums can be shown to approach 0. Although
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tomboi03
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The function f is defined on [0,1] so that f(1/n)=n^(-1/2) for n=1,2,3,... and f(x)=0 if x is not a reciprocal of a positive integer. Is f integrable on [0,1]? If so, prove it and compute the integral. If not then give an argument for why not.

See, I read this question over a hundred times, and the thing is... f(x) is always going to be a reciprocal of a positive integer... so the second statement is saying that any number inbetween the reciprocal of a positive integer, meaning all the irrational numbers inbetween 0 and 1 are equal to zero, no? Therefore this function is not continuous? Therefore this function cannot be integrable right? On top of that, if the function is 1 over a squareroot, then there will be two values for every reciprocal of a positive integer. (keeping in mind that a squareroot gives a positive and a negative answer right?

Am I wrong?

Thanks,
Jonnah Song
 
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  • #2
The function is integrable, Riemann or Lebesgue. Lebesgue is easy, since f=0 except on a set of measure 0. To show Riemann integrable, divide [0,1] into m subintervals, most of them will not contain a point where x=1/n. The upper Darboux sum can be seen to be < [sum(1,m) 1/n^1/2]/m ~ 1/m^1/2 -> 0. The lower Darboux sum is 0.
 
  • #3
sorry wrong statement, he said... "first notice that the function f is not continuous only at
countably many points."

sorry about that.
 

FAQ: Is f Integrable? Prove It or Give an Argument

1. What is the definition of an integrable function?

An integrable function is a function that can be represented as the area under a curve on a given interval. In other words, it is a function that can be integrated using the fundamental theorem of calculus.

2. How can I determine if a function is integrable?

A function is integrable if it is both continuous and bounded on a given interval. This means that the function has no breaks or discontinuities and its values do not approach infinity or negative infinity.

3. Can you give an example of an integrable function?

One example of an integrable function is f(x) = 2x + 1. This function is continuous and bounded on any interval, and its integral can be calculated using the power rule, resulting in the function F(x) = x^2 + x + C.

4. How do you prove that a function is integrable?

To prove that a function is integrable, you can use the Riemann sum method or the Darboux sum method. These methods involve dividing the interval into smaller subintervals and calculating the area under the curve for each subinterval. If the sums converge to a finite value, then the function is integrable.

5. What is the importance of determining if a function is integrable?

Determining if a function is integrable is important in many areas of mathematics and physics. It allows us to calculate the area under a curve, which has applications in finding volumes, centroids, and probabilities. It also helps us to understand the behavior and properties of functions.

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