Remark

Suppose that the space X is arcwise connected. Let $x$ be a point of X. For $f$ to be arcwise continuous, it suffices that, for every path $c$ with origin $x$, the mapping $f\circ c$ be continuous. Indeed, let $d$ be an arbitrary path in X and let $c$ be a path in X with origin $x$ and with term $d(0)$. If the mapping $f\circ (c*d)$ is continuous, the same is true of the mapping $f\circ d$ because we have $f\circ d(t) =f\circ (c*d)((t+ 1)/2)$.

A continuous mapping is arcwise continuous. The converse is not always true; the propositions 12 and 13 below provide criteria making it possible to assert that an arcwise continuous mapping is continuous.