I'm currently reading Ross's Elementary Analysis, which presents the definition of continuity as such: (not verbatim)(adsbygoogle = window.adsbygoogle || []).push({});

Let x be a point in the domain of f. If every sequence (x_{n}) in the domain of f that converges to x has the property that:

lim f(x_{n}) = f(x)

then we say that f is continuous at x.

For most functions, which are defined on uncountable sets, this presents no problems. But note that the sequences in question fall into two categories: those which converge "from the outside" (sequences in dom(f)\{x}) and those sequences which are eventually equal to x (and yes, I know that this does not actually form a partition of those sequences, but that won't really matter.)

So consider a function defined on some isolated point x. It may or may not be defined elsewhere, and the other places it is defined may or may not be isolated points, as long as none of them intersect some neighborhood of x. We've eliminated one of the two "types of convergent sequence." Every sequence in the domain that converges to x must eventually be equal to x, so f(x_{n}) eventually equals f(x), and the function is continuous at x, even though it's not even defined in the neighborhood of x.

The (equivalent) epsilon-delta formulation appears to allow for the same thing (obviously, or else it wouldn't be equivalent). Am I interpreting this correctly? Is it generally the case that we say such a sequence is continuous? Would it not make more sense to require the sequence in question to be in dom(f)\{x}? (Then again we rarely speak about continuity at a point anyways. All the interesting results concern continuity on a set, so I suppose it shouldn't affect much.)

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# Continuity in the metric spaces

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