Proposition 10

Let $X$ be a metrizable space and let $f$ be a mapping of $X$ into a topological space $X'$. Then $f$ is continuous at a point $x \in X$ if and only if, whenever $(x_n)$ is a sequence of points of $X$ which converges to $x$, the sequence $(f(x_n))$ converges to $f(x)$ in $X'$.

The condition is necessary, from Chapter I, § 7, no. 4, Proposition 9, Corollary 1. To show that it is sufficient, consider the filter $\mathscr{B}'$ of neighbourhoods of $f(a)$ in $X'$; the hypothesis implies that $\overline{f}^{-1}(\mathscr{B}')$ is coarser than every elementary filter associated with a sequence which converges to $a$, that is to say every elementary filter which converges to $a$; but

() See e.g., J. Dieudonné, Notes de Tératopologie (I), Revue scientifique (Revue rose), 1939, p. 39. (*) This proposition can still be valid in certain spaces in which at least one point does not have any countable fundamental system of neighbourhoods; for example, the space obtained by compactifying an uncountable discrete space by adjoining a point at infinity (Chapter I, § 9, no. 8, Theorem 4).

the intersection of these filters is the neighbourhood filter of $a$ (Chapter I, § 6, no. 8, Proposition 11). Hence the result.

Note that Propositions 8 and 10 are valid in any space $X$ in which every point has a countable fundamental system of neighbourhoods.