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Solved: The Dirichlet ruler function If x is a rational
Chapter 2, Problem 18AAE(choose chapter or problem)
PROBLEM 18AAE
The Dirichlet ruler function If x is a rational number, then x can be written in a unique way as a quotient of integers where n > 0 and m and n have no common factors greater than 1. (We say that such a fraction is in lowest terms. For example,6/4 written in lowest terms is 3/2) Let ƒ(x) be defined for all x in the interval [0, 1] by
a. Show that ƒ is discontinuous at every rational number in [0, 1].
b. Show that ƒ is continuous at every irrational number in [0, 1]. (Hint: If ∊ is a given positive number, show that there are only finitely many rational numbers r in [0, 1] such that
c. Sketch the graph of ƒ. Why do you think ƒ is called the “ruler function”?
Questions & Answers
QUESTION:
PROBLEM 18AAE
The Dirichlet ruler function If x is a rational number, then x can be written in a unique way as a quotient of integers where n > 0 and m and n have no common factors greater than 1. (We say that such a fraction is in lowest terms. For example,6/4 written in lowest terms is 3/2) Let ƒ(x) be defined for all x in the interval [0, 1] by
a. Show that ƒ is discontinuous at every rational number in [0, 1].
b. Show that ƒ is continuous at every irrational number in [0, 1]. (Hint: If ∊ is a given positive number, show that there are only finitely many rational numbers r in [0, 1] such that
c. Sketch the graph of ƒ. Why do you think ƒ is called the “ruler function”?
ANSWER:
Solution :
Step 1 :
In this problem, we have to The Dirichlet ruler function is discontinuous at every rational number.