Summary: 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.