Is the Function Continuous at Any Point?

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Homework Help Overview

The problem involves analyzing the continuity of a piecewise function defined on the real numbers, where the function takes different forms depending on whether the input is rational or irrational. Participants are tasked with identifying points of continuity for this function.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Some participants suggest examining specific points, such as -1 and 2, to determine continuity. Others propose using sequences of rational and irrational numbers to explore the function's behavior near these points.

Discussion Status

The discussion is ongoing, with participants sharing various thoughts on points of continuity and the methods to demonstrate them. Some guidance has been offered regarding the use of sequences to show continuity, but no consensus has been reached on the specific points where the function is continuous.

Contextual Notes

Participants are navigating the challenge of proving continuity at specific points, with some expressing uncertainty about how to approach the proof using sequences. There is also mention of a potential issue with missing information due to a lost proof attempt.

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Homework Statement



Let f:R->R be defined by f(x)=x^2 for x in Q and x+2 if x in R\Q. Find all points (if any) where f is continuous.


Homework Equations





The Attempt at a Solution

 
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f can only be continuous at 2.

Try showing this by using rational and irrational sequences approaching any point c.
 
Last edited:
how about -1?
but how to show it??
 
Use sequences of rationals number converging to an irrational number to show the function is not contituous at points other than -1,2.
 
Could you please help me with it? Please show me how this function is continuous at -1. Any help is appreciated..
 
f(-1) = 1 = (-1)^2 = (-1) + 2

All you have to do is show that both x^2 and x + 2 are arbitrarily close to 1 when x is sufficiently close to -1. This is quite simple because both of those functions are continuous at -1.

Formally, given \epsilon > 0, you need to find a \delta > 0 such that when |x - 1| < \delta, both x^2 and x + 2 are within \epsilon of 1. It's pretty easy to find such a \delta.
 
Last edited:
I wrote up a proof but it got eaten up by my browser.
Do you know how to proof continuity using sequences ?
 
Thank you all!
 
Did you figure out how to do the problem ?
 

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