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## Main Question or Discussion Point

please i am new to math. I donot know exact meanings of epsilon-delta definition. i dont comprehend it. Would Anybody help me. thanks in advance

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please i am new to math. I donot know exact meanings of epsilon-delta definition. i dont comprehend it. Would Anybody help me. thanks in advance

- #2

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In this case it reads:

A function [itex]f:D \rightarrow \mathbb{R}[/itex] is continuous at [itex]x_0 \in D[/itex] iff for any [itex]\epsilon>0[/itex] there exists a [itex]\delta>0[/itex] such that for all [itex]x \in D[/itex] with [itex]|x-x_0|<\delta[/itex]

[tex]|f(x)-f(x_0)|<\epsilon.[/tex]

This just says in a formal way that the graph of the function doesn't jump at [itex](x_0,f(x_0))[/itex].

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pwsnafu

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##0<|x-x_0|<\delta## actually.

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WannabeNewton

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https://www.physicsforums.com/showthread.php?t=700787please i am new to math. I donot know exact meanings of epsilon-delta definition. i dont comprehend it. Would Anybody help me. thanks in advance

- #5

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There's no reason to exclude the point ##x = x_0##. We trivially have ##|f(x_0) - f(x_0)| < \epsilon## for any ##\epsilon##.##0<|x-x_0|<\delta## actually.

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arildno

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It sure is a reason.There's no reason to exclude the point ##x = x_0##. We trivially have ##|f(x_0) - f(x_0)| < \epsilon## for any ##\epsilon##.

Otherwise, discontinuous functions would be deprived of limit values at the point of discontinuity. Thus, the limit concept would be conflated with the continuity concept.

Think about it!

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But vanhees71 was giving the definition of continuity at ##x_0##, not the definition of the existence of a limit at ##x_0##. If the function is to be continuous at ##x_0##, then it must be defined at ##x_0## and have the correct value!It sure is a reason.

Otherwise, discontinuous functions would be deprived of limit values at the point of discontinuity. Thus, the limit concept would be conflated with the continuity concept.

Think about it!

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arildno

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Hmm..no read it again.But vanhees71 was giving the definition of continuity at ##x_0##, not the definition of the existence of a limit at ##x_0##. If the function is to be continuous at ##x_0##, then it must be defined at ##x_0## and have the correct value!

What he posted was the definition in terms of the LIMIT concept in which L=f(x_0).

In particular, he writes d>0

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One more remark and then I'll shut up. :tongue: Here's a discussion from Spivak'sHmm..no read it again.

What he posted was the definition in terms of the LIMIT concept in which L=f(x_0).

In particular, he writes d>0

Spivak said:If we translate the equation ##\lim_{x \rightarrow a} f(x) = f(a)## according to the definition of limits, we obtain:

For every ##\epsilon > 0## there is a ##\delta > 0## such that, for all ##x##, if ##0 < |x - a| < \delta##, then ##|f(x) - f(a)| < \epsilon##.

But in this case, where the limit is ##f(a)##, the phrase ##0 < |x-a| < \delta## may be changed to the simpler condition ##|x-a| < \delta##, since if ##x = a## it is certainly true that ##|f(x) - f(a)| < \epsilon##.

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arildno

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Why change a perfectly good criterion for continuity?

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