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## Homework Statement

Let f(x) be deﬁned on [0,1] by

f(x) = 1 if x is rational

f(x) = 0 if x is irrational.

Is f integrable on [0,1]? You may use the fact that between any two rational numbers

there exists an irrational number, and between any two irrational numbers there exists

a rational number.

## Homework Equations

## The Attempt at a Solution

Divide into n sub-intervals.

Δx

_{i}=1/n

U(f,P

_{n}) = Ʃ(f(U

_{i})Δx

_{i}) = [itex]\sum(1)(1/n)[/itex] = 1/n

L(f,P

_{n}) = Ʃ[f(l

_{i})](Δx

_{i}) = [itex]\sum(0)(1/n)[/itex] = 0

As n[itex]\rightarrow[/itex] [itex]\infty[/itex] both U(f,P

_{n}) and L(f,P

_{n}) [itex]\rightarrow[/itex] 0

Therefore [itex]\int f(x)dx[/itex] = 0

Is this correct?

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