Discontinuous and continuous functions

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Discussion Overview

The discussion revolves around finding a function that is continuous at 0 but discontinuous at every other point. Participants explore various approaches and definitions related to continuity and limits.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Homework-related

Main Points Raised

  • One participant seeks a function that meets the criteria of being continuous at 0 and discontinuous elsewhere, expressing frustration over the challenge.
  • Another participant proposes a specific function defined piecewise, suggesting it may satisfy the required conditions.
  • Some participants question how a function can be continuous at only one point and still have a limit, prompting a discussion on the definitions of limits and continuity.
  • A participant reiterates the original question for clarity, emphasizing the need for an example of such a function.
  • There is a reference to the $(\varepsilon, \delta)$-definition of limits as a means to understand the continuity at a point.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the proposed function and the implications of continuity and limits, indicating that the discussion remains unresolved with multiple viewpoints presented.

Contextual Notes

Limitations include potential misunderstandings of the definitions of continuity and limits, as well as the challenge of finding a suitable example that meets the criteria.

Carla1985
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I need to find a function that is continuous at 0 but discontinuous at every other point. IV been stuck on this for hours now :( thankyou
 
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Re: discontinuous and continuous functions

Carla1985 said:
I need to find a function that is continuous at 0 but discontinuous at every other point. IV been stuck on this for hours now :( thankyou

Hey Carla! ;)

How about:
$$f(x) = \left\{\begin{aligned}
x & \text{ if } x \in \mathbb Q \\
-x & \text{ if } x \in \mathbb R \backslash \mathbb Q
\end{aligned}\right.$$
 
Carla1985 said:
I need to find a function that is continuous at 0 but discontinuous at every other point. IV been stuck on this for hours now :( thankyou

Continuous at just one point ! , then how does the limit exist ?
 
Not a clue. I don't get it at all. The exact wording of a question, just in case iv got it wrong is: "give an example of a function defined on R which is continuous at x=0 and discontinuous at every other point of R". I like Serena, thank you :)
 
ZaidAlyafey said:
Continuous at just one point ! , then how does the limit exist ?

Consider the definition of the limit of a function, using $(\varepsilon, \delta)$-definitions.
Combine it with the definition of a continuous function in a point.
 

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