Continuity of Dirichlet looking function

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Homework Help Overview

The discussion revolves around the continuity of a piecewise function defined as f(x) = x for rational x and f(x) = 0 for irrational x. Participants are exploring the conditions under which this function is continuous and examining the implications of limits approached through rational and irrational numbers.

Discussion Character

  • Conceptual clarification, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to analyze the continuity of the function by considering limits from rational and irrational sequences. They question whether their reasoning about the limits leading to different values is correct and seek further clarification on the concept of density in relation to the function's behavior.

Discussion Status

Some participants affirm the original poster's reasoning regarding the limits and continuity, while others engage in clarifying the concept of density and its implications for the function. There is an ongoing exploration of how to rigorously prove the density of rational and irrational numbers, with varying levels of technicality being discussed.

Contextual Notes

Participants express a desire to keep the discussion accessible, avoiding overly technical explanations beyond a second-year level of understanding. There is also a recognition of the need for clarity regarding definitions and proofs related to the density of numbers.

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Homework Statement



Where is the function f(x) continuous?

f(x) =
x, if x is rational

0, if x is irrational

Homework Equations





The Attempt at a Solution



Is this correct?: I approach some c =/= 0, 1st through x's that are rational
and prove there is the limit c, and then approach through x's that are irrational and prove that the limit now cannot be c, that now I can conclude that the limit at c does not exist, and hence the function is not continuous at any c=/=0?

If no, why, and in what other way must I solve it then? If yes, please try to explain as rigourously as you can why this can be done. Remind you though, don't get technical above 2nd year in which I am. Thank you.
 
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You have it right. If you approach through irrationals, the limit is always zero. Which is not equal to c unless c=0. There's not really much to get technical about beyond the statement that both the rationals and irrationals are dense in R.
 
Well you just did! Sorry, but by dense you mean that: for any pair of irrationals there is a rational between them, and the same the other way? This is what I vaguely remember I read some day on the web. How is it that dense implies you can do that? Thanks again.
 
A set S is dense if for any x you can find an s in S that is as close to x as you want. Which would let you say for any x you can find a sequence of rationals approaching it and a sequence of irrationals. Do you think you are expected to prove that?
 
No way! But how is it that the rationals and irrational are proven to be dense?
 
just basically showing that the limit as s in S approaches x is x itself, and that's it?
 
The proof depends on how you define an irrational number and can be sort of 'technical'. But try this, if x is rational, then {x+sqrt(2)/n} is a sequence of irrationals approaching x. If you think of an irrational in terms of it's decimal expansion then for example sqrt(2) is the limit of the sequence 1,1.4,1.41,1.414,... Each term in the series adds one more decimal place to the expansion and each term in the series is rational. How's that for 2nd year level?
 
Sweet! :) Even my neighbor can get that. I'll go there now and show it to him (joking). Thanks so much.
 

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