Functions of bounded \(\phi\)-variation appeared first in a paper of \textit{N. Wiener} [Massachusetts J. Math. 3, 72-94 (1924)]. Afterwards it was studied by others leading to generalizations and different perspectives. A \(\phi\)-function what is understood as far as this paper is concerned is a continuous, unbounded, non-decreasing function on \([0,\infty)\), with \(\phi (u)=0\) iff \(u=0\). Assuming \(F_ n\) to be finite-valued functions on \((-\infty,\infty)\), \(F_ n(0)=0,n=1,2,...\), and \(x\in {\mathcal V}_{\phi}[a,b]\), the class of functions of bounded \(\phi\)-variation, the authors' main result is to find a necessary and sufficient condition for the sequence \(Var_{\psi}(F_ n(x),a,b)\) (\(\psi\)-variation of \(F_ n(x)\) on \([a,b])\) to be bounded for each \(x\in {\mathcal V}_{\phi}[a,b]\) (where \(\psi\) is another \(\phi\)-function).