The class of functions of $\Lambda BV^{(p)}$ shares many properties of functions of bounded variation. Here we have shown that $\Lambda BV^{(p)}$ is a Banach space with a suitable norm, the intersection of $\Lambda BV^{(p)}$, over all sequences $\Lambda$, is the class of functions of BV$^{(p)}$ and the union of $\Lambda BV^{(p)}$, over all sequences $\Lambda$, is the class of functions having right- and left-hand limits at every point.