Dual space of KC

[Micael]

Dear all!

I am wondering if anything is known about the dual space of the
space of all functions $f: [0,1]\to R$, that have finite one-sided
limits in every point, endowed with $sup$-norm?
More exactly, let $M$ be the space of functions satisfying $f(x)=1/2 (f
(x+0)+f(x-0))$ at any $x\in (0,1)$ and $f(0+)=f(0)$, $f(1-)=f(1)$ with
$sup$-norm. Is there any description for $M*$?

Or, maybe, for the space of piecewise continuous functions on $[0,1]$?

I'd appreciate any reference/information!

Sincerely,
Michael.