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I don't understand the following:

Let $(X,d_x)$ and $(Y,d_y)$ be metric spaces, and let $f:X\to Y$ be a function.

Definition (of continuous function)

A function $f$ is called continuous at $x\in X$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that $d_y(f(x),f(y))<\varepsilon$ whenever $d(y,x) < δ$.

Definition (of limit of function) .. for every $\varepsilon > 0$ there exists $\delta > 0$ such that $d_y(f(x),l)<\varepsilon$ whenever $d(y,x) < δ$ $y\neq x$.

I don't really understand why we say that $y\neq x$ in the definition of limit of function and why we don't use $y\neq x$ in the definition of continuous function.

Thanks for your help!

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    $\begingroup$ Consider: if $f(x)=0$ when $x=0$ and $1$ otherwise, then $\lim_{x\to0} f(x)=1$ but $f$ is not continuous at $0$ $\endgroup$ – J. W. Tanner Mar 17 at 21:55
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    $\begingroup$ It might be helpful to point out that nothing as abstract and advanced as metric spaces is needed to understand what you're asking about, as the same issue arises when limits and continuity are introduced in a first course in calculus. Although the epsilon-delta definition of continuity may not have been formally covered in class, it's almost certainly in your calculus text somewhere. Possibly this answer might help in recalling what I'm talking about. $\endgroup$ – Dave L. Renfro Mar 17 at 22:03
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Continuity at $x$ is stronger than the existence of a limit at $x$.

Specifically, $f$ is continuous at $x$ if and only if

  • the limit of $f$ at $x$ exists and equals $l$, and
  • $l = f(x)$.

The first bullet point explains why "$l$" in the definition of limit is replaced by "$f(x)$" in the definition of continuity . The second bullet point explains why you need to remove the $y \ne x$ restriction.

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