# TIFR 2013 Problem 29 Solution -Continuity of function defined on rationals and irrationals

TIFR 2013 Problem 29 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programe leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert. This book is very useful for the preparation of TIFR Entrance.

Also Visit: College Mathematics Program of Cheenta

## Problem:True/False?

Let $f$ be a function on the closed interval $[0,1]$ defined by

$f(x)=x$ if $x$ is rational and $f(x)=x^2$ if $x$ is irrational.

Then $f$ is continuous at 0 and 1.

## Hint:

Use the sequence criterion for continuity.

## Discussion:

Let $x_n$ be a sequence converging to $0$. Then $x_n^2$ also converges to $0$. Since the value of $f$ at $x_n$ is either of the two, $f(x_n)\to 0$ as $n\to \infty$. If this seem confusing, think in terms of $\epsilon$. For $\epsilon >0$, there exists $n_1$ and $n_2$ such that $|x_n|< \epsilon$ and $|x_n^2|< \epsilon$ for $n>n_1$ and $n>n_2$ respectively. Taking the maximum of $n_1$ and $n_2$ we get the N for which the condition in "epsilon definition" is satisfied.

The same argument applies for the continuity at 1.

The function is not continuous at any other point. Because if a rational sequence and an irrational sequence converge to $x_0$ and the function is continuous at that point then by sequential criterion, $f(x_0)=x_0$ due to the rational sequence and also $f(x_0)=x_0^2$ due to the irrational sequence. Therefore, $x_0^2=x_0$ and hence the only possible points of continuity is $0$ or $1$.

## Helpdesk

• What is this topic: Real Analysis
• What are some of the associated concept: Continuous function, Closed Interval, Sequential criterion
• Book Suggestions: Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert

This site uses Akismet to reduce spam. Learn how your comment data is processed.

### Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.