# Ruler function proof

Define Ruler Function g:(0,1)-> R (it is the usual ruler function available online)

as g(x) = { 1/q if x = p/q, p and q are relatively prime

{ 0 if x is irrational.

Show (without using Riemann-Lebesgue Theorem) that g is Riemann integrable.

The hint says that since L (lower sum) is zero, it remains to show that U (Upper sum) is less than epsilon for some partition p. I'm trying to do that, but to me the answer is elusive.

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#### Solution Preview

Hello

Please find the solution in the attached files.

Proving Ruler Function is Riemann Integrable without using Riemann-Lebesgue Theorem

Define Ruler Function g:(0,1)-> R (it is the usual ruler function available online)

as g(x) = { 1/q if x = p/q, p and q are relatively ...

#### Solution Summary

This provides an example of proving the Ruler Function is Riemann Integrable without using Riemann-Lebesgue Theorem.