Riemann Integrability of f(x) = x on [0,1]

In summary, the conversation discusses the function f(x) and its integrability on the interval [0,1]. The speaker presents their reasoning for why the function is not Riemann integrable, citing the contradiction between two possible values for the integral. However, they also note that the function is Lebesque integrable, with a value of 0 due to the measure of the rational numbers. They also mention that a similar function with a different definition would have a Lebesque integral of 1/2.
  • #1
sihag
29
0
f(x) = x , if x is rational
= 0 , if x is irrational
on the interval [0,1]

i just wanted to check if my reasoning is right.

take the equipartition of n equal subintervals with choices of t_r's as r/n for each subinterval.

calculating the integral as limit of this sum (and sending the norm to 0) i got 1/2 as my value.

now if f were to be R - integrable the value of the integral must be 1/2.
but each subinterval for any partition would contain an irrational so the lower R sum would be 0, for all partitions of [0,1]
this yields 0 as the value of the integral.

the two values contradict.
 
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  • #2
Yes, if that function were integrable, then you would get two contradictory values. Conclusion: that function is not (Riemann) integrable.

(It is Lebesque integrable: since the rational numbers have measure 0, it Lebesque integral is 0. If the function were f(x)= x if is is irrational, 0 if x is rational, then its Lebesque integral would be 1/2.)
 

1. What is the Riemann Integrability of f(x) = x on [0,1]?

The Riemann Integrability of a function is a measure of how well a function can be approximated by a series of rectangles. For f(x) = x on [0,1], the Riemann Integral exists and is equal to 1/2.

2. How is the Riemann Integral of f(x) = x on [0,1] calculated?

The Riemann Integral of a function is calculated by partitioning the interval [0,1] into smaller subintervals and approximating the function with rectangles. The smaller the subintervals, the more accurate the approximation. The Riemann Integral is then defined as the limit of these approximations as the subintervals approach zero.

3. Can the Riemann Integral of f(x) = x on [0,1] be negative?

No, the Riemann Integral of a function on an interval is always a positive value. This is because the area under a curve is always a positive value, and the Riemann Integral measures this area.

4. What types of functions are Riemann Integrable?

A function is Riemann Integrable if it is continuous on the interval of integration and has a finite number of discontinuities. This means that the function must not have any sharp corners or vertical asymptotes within the interval.

5. How is the Riemann Integrability of a function related to its continuity?

The Riemann Integral exists for continuous functions, but not all continuous functions are Riemann Integrable. A function must also be bounded to be Riemann Integrable. This means that the function cannot approach infinity or negative infinity within the interval of integration.

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