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I don't know how to write "lim" with "x_{n}→ x" below on Microsoft Word's equation editor. And now I understand why my book has 0<|x-y|<∂ in the definition of continuity; I always assumed the "0<" part was irrelevant.

Perhaps you could try lim from {n rightarrow infty}?

Yes, that is one reason for 0<|x-y|<δ, which is necessary for derivatives.

In the case of a regular limit there is another reason though.

It is possible that the limit of a function is not equal to the function value in that point.

However, in that case the limit still exists.

So the definition of a limit in general requires that 0<|x-y|<δ.