Slight Discrepancy in Piece-wise Function Graph

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Discussion Overview

The discussion revolves around the properties of a piece-wise function, specifically addressing the concepts of limits and continuity at a point where the function has a defined value but differing one-sided limits. The scope includes mathematical reasoning and conceptual clarification related to calculus.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Brendan expresses confusion about the relationship between the defined value of the function at f(0) and the one-sided limits, questioning why f(0) is not equal to the limits.
  • Some participants emphasize the importance of distinguishing between limits and function values, suggesting that the exercise should not pose a problem if these concepts are understood correctly.
  • Brendan raises a question about the continuity of the function at x = 0, noting that the limits differ from the defined value, which complicates the classification of continuity.
  • One participant asserts that x = 0 represents a discontinuity for the function, while Brendan seeks clarity on how to respond to a multiple-choice question regarding continuity when none of the options seem applicable.
  • Another participant suggests that "NEITHER" could be a valid answer to the multiple-choice question, indicating that the provided options may not encompass all possibilities.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the limits and the defined function value, with some agreeing on the discontinuity at x = 0 while others debate the appropriateness of the multiple-choice options given the situation. The discussion remains unresolved regarding the best approach to answer the quiz question.

Contextual Notes

The discussion highlights limitations in the definitions and options provided in the quiz, as well as the potential for misunderstanding the relationship between limits and function values. There is an unresolved tension regarding the classification of continuity based on the given information.

brendan_foo
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Hi there,

Only a slight problem here... My girlfriend has just taken a calc 1 quiz and she's presented this problem to me which goes against the grain of what I know - however, I could be hideously mis-informed.

She was asked to draw a graph of a piece-wise function given some definitions and limits. I shall only state the area that is causing me some intuitive grief.

[tex] f(0) = 2[/tex]
[tex] \lim_{x\to{0^-}} f(x) = -1[/tex]
[tex] \lim_{x\to{0^+}} f(x) = 1[/tex]

If f(0) is most definitely defined at that point, why is it not its limit. By this logic, couldn't i essentially dot defined, singular points all over any arbitrary graph with DEFINITE two sided limits...This is my problem...

Any opinions?

Cheers guys...Brendan
 
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Well, as long as you don't confuse the concepts of limits and function values, this exercise cannot possibly represent any problems.
 
This is what I argued, but wasn't received with much gratitude.

Thankyou for this...

Cheers...Brendan
 
Okay, remember that in order for a one-sided limit to exist at some point, then the limiting function value of EACH SEQUENCE OF NUMBERS CONVERGING TO THAT POINT (from the one side) must equal the limiting function value associated with every other sequence.
 
Quick other point:

It then asks...is the function left continuous, right continuous or continuous at the point x = 0. Based on the definition of continuity, that is,

[tex] \lim_{x\to a}f(x) = f(a)[/tex]

Based on the two stipulated limits (as is posted above), what could I say in this case, as the limits are defined as different to the stated existing point [itex]f(0) = 2[/itex].

Cheers...Brendan
 
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x=0 is a screaming discontinuity for this function
 
Again, this is what I argued - however, as a multiple choice question as: left cont, right cont, or continuous in both senses...how is one to answer?! This is the point that I make.
 
It is none of them. Just look at the definitions you have been given.
 
I can 100% see this; but when faced with a) b) or c) and nothing in between, it seemed a little peculiar, especially with her screaming down my neck that it MUST be one of the three, even though it clearly isn't.

Thanks guys, just wanted to get some confirmation before I got on my high-horse.

Mucho-gracias!
 
  • #10
Well, as you actually PHRASED it (without the a), b) and c)), NEITHER is a perfectly acceptable answer!

If the book clearly has stated your three only options as the ones you mentioned, I suggest you make a cozy fire of it.
 
  • #11
I quote verbatim from the cherished one... It wasn't a book exercise, it was a quiz devised by some one.
 
  • #12
Yes, so as I said "NEITHER" is a perfectly acceptable answer.
 

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