On Some Properties of Separately Increasing Functions from $[0,1]^n$ into a Banach Space

We say that a function $f$ from $[0,1]$ to a Banach space $X$ is increasing with respect to $E\subset X^*$ if $x^*\circ f$ is increasing for every $x^*\in E$. A function $f:[0,1]^m\to X$ is separately increasing if it is increasing in each variable separately. We show that if $X$ is a Banach space that does not contain any isomorphic copy of $c_0$ or such that $X^*$ is separable, then for every separately increasing function $f:[0,1]^m\to X$ with respect to any norming subset there exists a separately increasing function $g:[0,1]^m\to \mathbb R$ such that the sets of points of discontinuity of $f$ and $g$ coincide.