$K$-to-$1$ functions on arcs for $K$ even

For exactly $k$-to-$1$ functions from $[0,1]$ into $[0,1]$:

(1) at least one discontinuity is required (Harrold),

(2) if $k = 2$, then infinitely many discontinuities are needed, for any Hausdorff image space (Heath),

(3) if $k = 4$, or if $k$ is odd, then there is such a function with only one discontinuity (Katsuura and Kellum), and, it is shown here that

(4) if $k$ is even and $k > 4$, then there is such a function with only two discontinuities, and no such function exists with fewer discontinuities.