Proposition 12

Let X and Y be topological spaces and let $f: X\rightarrow Y$ be a mapping. Suppose that the space X is locally arcwise connected and that every point of X possesses a countable fundamental system of neighbourhoods. If the mapping $f$ is arcwise continuous, it is continuous.

According to (TG, IX, p. 18), it suffices to prove that, for every point $x$ of X and every sequence $(x_n)_{n\geqslant 1}$ of points of X which tends to $x$, the sequence $(f(x_n))_{n\geqslant 1}$ tends to $f(x)$. By deleting, if necessary, the first terms of the sequence $(x_n)_{n\geqslant 1}$, we reduce to the case where all the terms of the sequence belong to the same arc-connected neighbourhood of $x$ in X. According to the lemma below, there then exists a path $c:\mathbf{I}\rightarrow X$ such that $c(0) =x$ and $c(1/n) =x_n$ for $n\geqslant 1$. If the mapping $f$ is continuous by arcs, the mapping $f\circ c$ is continuous and the element $f(x) =f(c(0))$ is the limit of the sequence $(f(c(1/n)))_{n\geqslant 1}$, that is to say of the sequence $(f(x_n))_{n\geqslant 1}$, whence the corollary.