How do you prove Dirichlet function is not Riemann integrable?
To prove that Dirichlet function is not Riemann integrable you have to show that Upper Riemann sum and Lower Riemann Sum do not converge to the same number when you make partition thin and thin of the interval on which function is defined or in other words upper Riemann integral is not equal to lower Riemann integral.
Since Dirichlet function is defined as
If
1 - X belong to rational numbers then f(x) = 1
2 - X belong to irrational numbers then f(X) = 0
Let Dirichlet function is defined on [a , b] and P ={ a=x_1, x_2,...,x_n=b } be Partition of [a,b].
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| Is Dirichlet function Riemann integrable? |
Since for above function
M_ r = sup f(X)=1 & m _ r = inf f(X)=0
This implies that
U(P , f) = ∑ Mrδr where δr = x_n -x_1
U(P, f) = ∑ (1) δr = x_n - x_1 = b – a
&
L(P, f) =∑ mrδr where δr = x_n -x_1
L(P, f)= =∑ (0) δr = 0 (x_n - x_1) = 0 (b - a ) = 0
This implies that
Upper Riemann integral= U(f) = inf U(P, f)
U(f) = inf U(P, f) = inf (b – a) = b - a
&
Lower Riemann integral= L(f) = sup L(P, f)
L(f) = sup L(P, f) = sup (0) = 0
Therefore
b – a ≠ 0
U(f) ≠ L(f)
Where
U(f) = Upper Riemann Integral
L(f) Lower Riemann integrable
Hence proved that Dirichlet function is not Riemann integrable because upper Riemann integral is not equal to lower Riemann integral.
Frequently Asked Questions
Is Dirichlet function Riemann integrable?
No Dirichlet function is not Riemann integrable.
Why is the Dirichlet function not the Riemann integrable?
Dirichlet function is not Riemann integrable because it has infinite points of discontinuity on any interval.
Is Dirichlet function continuous?
No Dirichlet function is not continuous.