Cheenta
How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

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

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com