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# Ruler function

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5. Every rational x can be written in the form x = m/n, where n > 0, and m and n are integers without any common divisors. When x = 0, we take n = 1. Consider the function f defined on the reals where f(x) = 0 when x is irrational and f(x)= 1/n when x =m/n

a, prove that f(x) is continuous at every irrational point and discontinuous at every rational point
b, show that f is integrable on [0,1]

https://brainmass.com/math/graphs-and-functions/ruler-function-345008

#### Solution Preview

5. Every rational x can be written in the form x = m/n, where n > 0, and m and n are integers without any common divisors. When x = 0, we take n = 1. Consider the function f defined on the reals where f(x) = 0 when x is irrational and f(x)= 1/n when x =m/n

a, prove that f(x) is continuous at every irrational point and discontinuous at every rational point
b, show that f is integrable on [0,1]

a) Let be a rational number so that . Since in every neighborhood of rational ...

#### Solution Summary

Ruler function is resolved.

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