An investigation of Heine´s characterization of continuity

Question

This characterization of continuity, for example, for functions $f: \mathbb R \to \mathbb R$, can be stated as:

A function $f: \mathbb R \to \mathbb R$ is continuous at the point $x_0$ if and only if for every sequence $x_k$ that tends to $x_0$ the sequence $f(x_k)$ tends to $f(x_0)$.

I am trying to minimize the assumptions on sequences as much as possible. for example, I think that Heine´s characterization of continuity would be valid if we would phrase it like this:

A function $f: \mathbb R \to \mathbb R$ is continuous at the point $x_0$ if and only if for almost all sequences $x_k$ that tend to $x_0$ the sequences $f(x_k)$ tend to $f(x_0)$.

Now, the meaning of "almost all" can, I believe, be somehow given a more precise meaning.

I am not sure would this phrasing be enough to secure continuity:

A function $f: \mathbb R \to \mathbb R$ is continuous at the point $x_0$ if and only if for all (except maybe countably many) sequences $x_k$ that tend to $x_0$ the sequences $f(x_k)$ tend to $f(x_0)$.

What do you think about this, in the sense of minimization of assumptions? That is, how much can a requirement on all sequences be weakened so as to still have this theorem valid?

Edit: To clarify, I mean that if we suppose that continuity is preserved for some space of sequences smaller than the space of all sequences, then we can show that it is preserved for all sequences. But, how "small" is small? Everywhere dense set of sequences?

Answer 1

Your weakening to all but possibly countably-many sequences works just perfectly. To prove it, we need only prove the following

Claim: Suppose $f:\Bbb R\to\Bbb R$, and that there is a sequence of points $x_k$ of $\Bbb R$ and a point $x_0\in\Bbb R$ such that $x_k\to x_0$ but $\neg\bigl(f(x_k)\to f(x_0)\bigr).$ Then there are uncounably-many sequences of points $y_k$ of $\Bbb R$ such that $y_k\to x_0$ but $\neg\bigl(f(y_k)\to f(x_0)\bigr).$

The proof is easy: Simply changing $x_1$ to any other point in $\Bbb R$ allows us to construct uncountably-many such sequences--in fact, continuum-many sequences!

Thus, the best we could do is specify that the number of sequences for which the convergence condition may fail must be less than continuum-many. This may actually be equivalent to saying that the number must be at most countably-many, but it is undecidable in basic set theory whether there are cardinalities between $\aleph_0$ (the cardinality of the natural numbers) and $\mathfrak{c}=\mathbf{2}^{\aleph_0}$ (the cardinality of the set of real numbers and of the power set of the natural numbers).