Continuity Problem: Solutions & Resources

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Discussion Overview

The discussion revolves around the concept of continuity of a function, specifically examining the conditions under which a function is continuous at a point, with a focus on the point \(x=1\). Participants explore the limits and function values, as well as continuity at other points.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant states that a function is continuous at a point if its limit at that point equals its function value.
  • Another participant claims to find both \(f(1)\) and \(\lim_{x \to 1} f(x) = 0\) from both the left-hand limit (lhl) and right-hand limit (rhl).
  • A subsequent participant reiterates the previous findings about \(f(1)\) and the limits, concluding that the function is continuous at \(x=1\) and suggesting that answer option a has been eliminated.
  • This participant raises further questions about the continuity of the function at other integer values of \(x\) and whether there exists a real \(x\) where the function is not continuous.

Areas of Agreement / Disagreement

Participants appear to agree on the continuity of the function at \(x=1\) based on the limits and function value discussed. However, the exploration of continuity at other points remains unresolved, with multiple questions posed about the function's behavior elsewhere.

Contextual Notes

The discussion does not clarify the specific form of the function \(f\), which may affect the analysis of continuity at other points. There are also unresolved questions regarding the continuity at integer and real values of \(x\) beyond \(x=1\).

Who May Find This Useful

Students or individuals studying calculus, particularly those interested in the properties of continuity and limits of functions.

DaalChawal
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A function is continuous at a point if its limit at that point is the same as its function value.

What is $\lim_{x\to 1} f(x)$ and what is $f(1)$?
 
I'm getting $f(1)$ as well as $\lim_{x \to 1}f(x)$ = 0 both lhl and rhl
 
DaalChawal said:
I'm getting $f(1)$ as well as $\lim_{x \to 1}f(x)$ = 0 both lhl and rhl
Then $f$ is continuous at $x=1$ and we have eliminated answer a.

The remaining question is what happens at other values of $x$.
Can we find another integer value for $x$ where $f$ is continuous?
Can we find a real $x$ where $f$ is not continuous?
 

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