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 \(x^*\in E\). We show that if \(f:[0,1]\to X\) is an increasing function with respect to a norming subset \(E\) of \(X^*\) with uncountably many points of discontinuity and \(Q\) is a countable dense subset of \([0,1]\), then
(1) \(\overline{\text{lin}\{f([0,1])\}}\) contains an order isomorphic copy of \(D(0,1)\),
(2) \(\overline{\text{lin}\{f(Q)\}}\) contains an isomorphic copy of \(C([0,1])\),
(3) \(\overline{\text{lin}\{f([0,1])\}}/\overline{\text{lin}\{f(Q)\}}\) contains an isomorphic copy of \(c_0({\varGamma})\) for some uncountable set \({\varGamma}\),
(4) if \(I\) is an isomorphic embedding of \(\overline{\text{lin}\{f([0,1])\}}\) into a Banach space \(Z\), then no separable complemented subspace of \(Z\) contains \(I(\overline{\text{lin}\{f(Q)\}})\).