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# ISI MStat 2018 PSA Problem 10 | Dirichlet Function This is a problem from ISI MStat 2018 PSA Problem 10 based on Dirichlet Function.

## Dirichlet Function - ISI MStat Year 2018 PSA Question 10

Let be a real number. Then • (A) does not exist for any x
• (B) exists for all x
• (C) exists if and only if x is irrational
• (D) exists if and only if x is rational

### Key Concepts

Limit

Sandwich Theorem

ISI MStat 2018 PSA Problem 10

Introduction to Real Analysis by Bertle Sherbert

## Try with Hints

Check two cases separately one when x is rational and other is when x is irrational.

If is an integer, then If x is rational , then, eventually, for large enough m, m! will be divisible by q , so that will be an integer, and we have If x is irrational, will never be an integer, and , so that for all m>0 by Sandwich Theorem.

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This is a problem from ISI MStat 2018 PSA Problem 10 based on Dirichlet Function.

## Dirichlet Function - ISI MStat Year 2018 PSA Question 10

Let be a real number. Then • (A) does not exist for any x
• (B) exists for all x
• (C) exists if and only if x is irrational
• (D) exists if and only if x is rational

### Key Concepts

Limit

Sandwich Theorem

ISI MStat 2018 PSA Problem 10

Introduction to Real Analysis by Bertle Sherbert

## Try with Hints

Check two cases separately one when x is rational and other is when x is irrational.

If is an integer, then If x is rational , then, eventually, for large enough m, m! will be divisible by q , so that will be an integer, and we have If x is irrational, will never be an integer, and , so that for all m>0 by Sandwich Theorem.

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### 2 comments on “ISI MStat 2018 PSA Problem 10 | Dirichlet Function”

1. Arnab Chattopadhyay. says:

What I can see that the limit exists for all real If is rational the limit evaluates to and if is irrational the limit evaluates to Then why do say that the limit doesn't exist for any real ? The reasoning is not quite clear to me. Can you please explain?

1. Pravat Hati says:

Don't worry. It will be (B) i.e limit exists for all real x we have also shown that . I mistakenly put A as option . Between thanks for your valuable reply.

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