$k$-to-$1$ functions on an arc

Recently Jo W. Heath [6] has shown that any $2$-to-$1$ function from an arc onto a Hausdorff space must have infinitely many discontinuities. Here we investigate extending Heath's result to $k$-to-$1$ functions for $k > 2$. Examples show that in general Heath's theorem cannot be extended even for functions from an arc into itself. However, if $f$ is a $k$-to-$1$ function $(k \geq 2)$ from an arc onto an arc, then we prove that $f$ has infinitely many discontinuities.