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ABSTRACT. Recently Jo W. Heath [6] has shown that any 2-to-l function from an arc onto a Hausdorff space must have infinitely many discontinuities. Here we investigate extending Heath's result to fc-to-1 functions for fc > 2. Examples show that in general Heath's theorem cannot be extended even for functions from an arc into itself. However, if / is a fc-to-1 function (fc > 2) from an arc onto an arc, then we prove that / has infinitely many discontinuities. Introduction.A function / is called fc-to-1 for some integer fc if f~l(y) contains exactly fc points for each y in the image of /. Since Harrold Here we investigate extending Heath's result to fc-to-1 functions for fc > 2. It follows from Harrold [5] that no continuous fc-to-1 function exists from an arc to an arc for fc > 2. Examples show that in general Heath's theorem cannot be extended even for functions from an arc into itself. However, if / is a fc-to-1 function (fc > 2) from an arc onto an arc, then we prove that / has infinitely many discontinuities.Our proofs are elementary, making use

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