Show this silly function is Riemann-integrable and find the integral

In summary, the conversation is about proving the integrability of a function that is defined as 0 for irrational numbers and 1/q for rational numbers of the form p/q where p and q are coprime. The main question is how to show that the set of discontinuities, which is assumed to be the rationals in the interval [0,1], has measure zero. One approach suggested is to show that for any irrational number y and a given margin of error e, there exists a disk around y such that all rationals in the disk have a denominator less than 1/e. This can be achieved by choosing a finite set of rationals with denominators less than 1/e and taking the infimum of the
  • #1
quasar987
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Homework Statement


See the title. The silly function in question is f:[0,1]-->R with f(x)=0 if x is irrational, and f(x)=1/q if x is rational and of the form x=p/q where p and q have no common factor.

Homework Equations


The Attempt at a Solution



I'm like 100% sure that I must show it is integrable by showing that the set of discontinuities is of measure zero and the natural assumption is that this set of discontinuities is the rationals in [0,1], but how do I show that?

I feel there is something I am missing about the p/q representation thing. If f is continuous on the irrational, then it must be that given an e>0 and an irrational y, there is a little disk around it y such that all rationals in that disk have 1/q<e. How come?
 
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  • #2
How many rationals are there in an interval?
 
  • #3
Ok, if y is irrational and you are given e>0, let Q be an integer such that Q>1/e. Now let S be the set of rationals {p/q} where q<=Q. S is a finite set. So let the radius of the disk be delta=inf(|y-S|)>0.
 
  • #4
That is ingenious!
 

FAQ: Show this silly function is Riemann-integrable and find the integral

What is a Riemann-integrable function?

A Riemann-integrable function is a mathematical function that can be integrated using the Riemann integral. This means that the area under the curve of the function can be calculated using a specific method called the Riemann sum.

How do you show that a function is Riemann-integrable?

In order to show that a function is Riemann-integrable, you must demonstrate that the upper and lower Riemann sums of the function converge to the same value. This can be done by dividing the area under the curve into smaller and smaller rectangles and calculating the sum of their areas.

What is the process for finding the integral of a Riemann-integrable function?

The process for finding the integral of a Riemann-integrable function involves dividing the area under the curve into smaller and smaller rectangles, calculating the sum of their areas, and then taking the limit as the number of rectangles approaches infinity. This limit represents the integral of the function.

Are there any properties or characteristics that Riemann-integrable functions share?

Yes, there are several properties and characteristics that Riemann-integrable functions share. These include being bounded, having a finite number of discontinuities, and being continuous almost everywhere.

Why is it important to determine if a function is Riemann-integrable?

Determining if a function is Riemann-integrable is important because it allows us to calculate the area under the curve of the function, which has many practical applications in fields such as physics, engineering, and economics. It also helps us understand the behavior and properties of the function in a mathematical sense.

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