Is the Limit of a Continuous Function Equal to the Limit of its Variable?

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Discussion Overview

The discussion centers around the relationship between the limit of a continuous function and the limit of its variable as it approaches a specific point. Participants explore whether the limit of a continuous function at a point equals the function's value at that point, particularly in the context of continuity.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants assert that if a function f is continuous at x = a, then the limit as x approaches a of f(x) equals f(a), based on the definition of continuity.
  • Others propose that the expression \lim_{x \rightarrow a} f(x) = f( \lim_{x \rightarrow a} x) is a valid way to express continuity, as the limit of x as it approaches a is simply a.
  • A participant questions the phrasing of the limit expression but acknowledges that it aligns with the definition of continuity.
  • Another participant confirms that the limit expression is true and emphasizes that it clarifies the concept of continuity.

Areas of Agreement / Disagreement

Participants generally agree on the relationship between the limit of a continuous function and its value at that point, but there is some variation in how they express and interpret this relationship.

Contextual Notes

The discussion does not resolve potential ambiguities in the phrasing of the limit expression or the implications of continuity in different contexts.

JG89
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If f is continuous in some neighborhood of x = a, then is the following true:

[tex]\lim_{x \rightarrow a} f(x) = f( \lim_{x \rightarrow a} x)[/tex]?
 
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If f is continuous in some neighborhood of x = a, it is also continuous at x = a, because x = a is contained in the neighborhood. The l.h.s equals the r.h.s because of the fact that the limit as x tends to 'a' of f(x) equals 'f(a)' (because of continuity of f) and on the other hand, f of the limit of x as 'x tends to a' is obviously f(a) since lim(x) = a as x --> a
 
Last edited:
JG89 said:
If f is continuous in some neighborhood of x = a, then is the following true:

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

This an anorthodoxe way of writting : f is continuous at x=a <====>
[tex]\lim_{x \rightarrow a} f(x)=f(a)[/tex]

But i suppose is correct since [tex]\lim_{x\rightarrow a}x = a[/tex]
 
JG89 said:
If f is continuous in some neighborhood of x = a, then is the following true:

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

That is true. This makes clear the idea of continuity.
 

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