
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
May 15th, 2017, 06:41 PM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 148 Thanks: 1  Number of points of continuity
The number of points of continuity of the function f = $ x^2  1$ if $x$ is irrational, $0$ if $x$ is rational A) $0$. B) $1$. C) $2$. D) Infinite I hope if you substitute irrational numbers in the function we get a rational answer because of the "square". So the function is continuous at infinite points. If I'm wrong correct me please, thank you 
May 15th, 2017, 07:09 PM  #2 
Senior Member Joined: Aug 2012 Posts: 1,370 Thanks: 321 
This depends on $f$. In any event $\pi^2$ is irrational. Or is the expression the function? Not clear.

May 15th, 2017, 07:21 PM  #3 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 148 Thanks: 1  
May 15th, 2017, 08:34 PM  #4 
Senior Member Joined: Aug 2012 Posts: 1,370 Thanks: 321 
Oh right. For some reason I read that as "the number of points of continuity" of the function was that many. So we have this: $$ f(x) = \left\{ \begin{array}{ll} 0 & x \in \mathbb Q \\  x^2  1 & x \notin \mathbb Q \end{array} \right. $$ Let's see. The first thing is that it's zero on the rationals and we know that a continuous function is determined by its values on the rationals. For example if it's continuous at $\pi$ then since the function is $0$ on each of $3, 3.1, 3.14, \dots$ it must be $0$ at $\pi$. But $\pi^2  1 \neq 0$ so $f$ can't be continuous at any irrationals. The same argument applies to rationals in fact. Every rational is also the limit of a sequence of rationals so if $f$ is continuous at a rational it must be $0$ there too. So the only values where the function can be $0$ and also $x^2  1 = 0$ are $\pm 1$. I don't know why they are confusing the issue with the absolute value. So I would go with C) 2. 
May 15th, 2017, 10:07 PM  #5  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 148 Thanks: 1  Quote:
I understood it. Thank you so much for the explanation  

Tags 
continuity, number, points 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
the total number of points in ndimensional sphere  Business Man  Algebra  7  October 24th, 2013 06:19 PM 
Continuity of functions in isolated points  galileo  Real Analysis  2  September 16th, 2012 01:39 PM 
The number of points  skarface  Algebra  3  March 11th, 2012 12:32 PM 
Number of Points of intersection.  chetjan  Algebra  11  November 6th, 2009 02:11 PM 
Number of integration points in Gauss quadrature  Luc4  Applied Math  1  September 4th, 2009 09:32 AM 