The spaces characterized by the property that a subset is closed if and only if it is closed w.r.t. limits of transfinite sequences are called pseudoradial spaces.
I would like to give an example of a space that is not pseudoradial together with sketch of a proof. (I hope I am not doing something very easy in a too complicated way.) I would also like to mention a few references and some properties of these spaces.
Before presenting the example, one short remark. Pseudoradial spaces are represented by the convergence of net on well-ordered nets. Obviously, it is sufficient to take the nets on ordinals. We can go a little further - regular cardinals are sufficient. Indeed, if we have a cofinal subset of an ordinal, we can use this cofinal subset to get another convergent net.
Let us consider the following example. Each arrow in the picture bellow represents a convergent sequence. I.e., this is a topological space homeomorphic to $\{0\}\cup\{\frac1n;n\in\mathbb N\}$ taken as a subspace of real line. Equivalently, this is precisely the ordinal $\omega+1$ taken with the order topology.
We take all these sequences and identify some of the points as in the picture. (I.e., we make a quotient space of some of these spaces.) Let us call the resulting space $S_2$. Then we take the subspace of this space as shown in the picture. This subspace will be called $S_2^-$. (I've taken the notation $S_2$ and $S_2^-$ from this paper: Franklin S.P., Rajagopalan M., On subsequential spaces, Topology Appl. 35 (1990), 1-19. But you can notice that this space is very very similar to Arens-Fort space mentioned in Brian's answer.)
Now we want to show that $S_2^-$ is not pseudoradial.
Note that the space $S_2^-$ has only one non-isolated point. Let us call it $\omega$ . So we ask whether there is a transfinite sequence, consisting only of points different from $\omega$, which converges to $\omega$.
First, let us show that this is not possible for a regular cardinal $\alpha>\omega$. Suppose that $(x_\eta)_{\eta<\alpha}$ is an $\alpha$-sequence of points of $S_2^-\setminus\{\infty\}$, which converges to $\infty$. Let us denote $n_\eta$ the "column" to which $x_\eta$ belongs. In we use the notation the notation from the picture bellow $n_\eta$ is the first coordinate of ordered pair $x_\eta$.
We can see that $n_\eta$ converges to $\omega$. (E.g. by noticing that $(x,y)\mapsto x$ and $\omega\to\omega$ is a quotient map from $S_2^-$ to $\omega$ with order topology.)
Now this is not possible, since the we would be able to construct an increasing $\alpha$-sequence converging to $\omega$ and using this sequence we would be able to show that cofinality of $\alpha$ is $\omega$.
So the only possibility is to take a sequence in the usual sense, i.e., a sequence of length $\omega$. Perhaps with a little handwaving, but it is more-or-less clear that general situation is similar to the situation when the $n$-term of the sequence is in the $n$-th column. So we have a sequence $x_n=(n,y_n)$. Obviously $\{\omega\}\cup\bigcup\limits_{n\in\omega} \{n\}\times(y_n,\infty)$ is a neighborhood of the point $\omega$ containing no terms of this sequence.
Pseudoradial spaces were introduced by H. Herrlich. Quotienten geordneten Räume und Folgenkonvergenz. Fund. Math., 61:79–81, 1967; pdf. They were later studied by A.V. Arhangelskii and many others.
The class of pseudoradial spaces is closed under the formation of closed subspaces, quotients and topological sums. They are a coreflective subcategory of the category Top of all topological spaces. This means that for each topological space we have pseudoradial coreflection; a pseudoradial space which is, in some sense, close to this space. The pseudoradial coreflection is obtained simply by taking sets closed under limits of transfinite sequences as closed sets in a new topology on the same set. (E.g. the pseudoradial coreflection of $S_2^-$ is discrete.)
The same thing can be done with any class $\mathbb P$ of directed sets instead of ordinals. This is called $\mathbb P$-net spaces in P. J. Nyikos. Convergence in topology. (In M. Hušek and J. van Mill, editors, Recent Progress in General Topology, pages 537–570, Amsterdam
1992. North-Holland.) The properties of pseudoradial spaces which I mentioned in the preceding paragraph are true for $\mathbb P$-net spaces, too.
Interestingly, if we take the linearly ordered sets, we obtain the same class of spaces as from well-ordered sets, see
James R. Boone: A note on linearly ordered net spaces. Pacific J. Math. Volume 98, Number 1 (1982), 25-35; link.
Kelley General Topology (p. 74, Thm 9) mentions the following 4 conditions on net convergence to define a "convergence class" (on a set $X$):
a. If $S$ is a net such that $S_n = s \in X$ for each $n$ ($n \in N$, some directed set), then $S$ converges to $s$.
b. If $S$ converges to $s \in X$, then so does every subnet of $S$ (subnet in Kelley's sense, of course, defined on p. 70 of the book).
c. If $S$ does not converge to $s \in X$, then there is a subnet of $S$ no subnet of which converges to $s$. (A condition that a.e. convergence fails to fulfill, as shown by this argument by Ordman).
d. (what he refers to as theorem 2.4 on iterated limits): Let $D$ be a directed set, let $E_m$ be a directed set for each $m \in D$, let $F= D \times \prod_{m \in D} E_m$ (in the product (i.e. coordinatewise) ordering, which is directed too) and for each $(m,f) \in F$ let $R(m,f)=(m,f(m))$. If $\lim_m \lim_n S(m,n)=s$ ($S$ is a function into $X$ defined on all pairs $(m,n)$ with $m \in D \land n \in E_m$) for some $s \in X$ (according to the "convergence rule"), then $S \circ R$ converges to $s$ too.
The last condition is a technical one to ensure that the set of limits of nets from a set $A$, will be a valid idempotent ($\overline{\overline{A}} = \overline{A}$) closure operator. Read the book for details. All conditions a-d are valid in a topological space convergence, and if a convergence class obeys them we can define a topology for which this convergence is the topology-defined convergence.
Best Answer
This is dependent on whether the topology on $X$ is so-called sequential, i.e. can be described completely using convergent sequences.
A set $A \subseteq X$ is called sequentially closed iff for all sequences $(a_n)_n$ in $A$ (i.e. $a_n \in A$ for all $n \in \Bbb N$) such that $a_n \to x$ in $X$ we can conclude that $x \in A$ as well.
In every topological space $X$ a closed subset is sequentially closed but if the converse holds (every sequentially closed $A$ is closed in $X$) then $X$ is called a sequential space. So we only need to know what sequences converge in $X$ to know what all closed subsets (and hence all open subsets too) in $X$ are. Most common spaces are sequential, i.e. all first countable (with a countable local base at every point) spaces are sequential, so all metric spaces are. This "explains" why sequence arguments are so common in analysis (where we mostly work with metric spaces) and sufffice to show continuity etc.
A simple to describe space that is not sequential is the co-countable topology on $\Bbb R$ (a set $A$ is closed iff $A=\Bbb R$ or $A$ is at most countable), and another is $\Bbb R^I$ where $I$ is of cardinality continuum. Some more (Cech-Stone compactifications e.g.) exist. This motivates the introduction of nets where we can show in any space that closed="closed under net-limits" and continuous = "net-continuous", etc.
So $X$ being sequential is key for your continuity.