Most of the non-linear problems occurring in physical and engineering aspects have generalized solutions via functions of bounded variation by forming an algebra of discontinuous functions. However, sometimes there is a need to analyze the functions that are of unbounded variation that arise naturally in medical and engineering phenomena. One such work has been done by dealing with a new continuous function of unbounded variation. Its characterizations are analyzed on the closed unit interval [0, 1]. A powerful tool to determine the order of non-linear auto-regressive model, to distinguish between random noise and to test the residuals of linear models is the fractal dimension. Many fractal models as said above can be characterized by their fractal dimension. Thus, this work incorporates a discussion on the fractal dimension of the graph of (k, s)-Riemann–Liouville fractional integral of the novel continuous function on [0, 1] and it is predicted that both the fractal dimensions are 1. Additionally, the other fractal dimensions for example packing dimension and K-dimension for both graphs have been calculated. Finally, the connection between the fractal dimension of (k, s)-Riemann-Liouville fractional integral and its order is shown as the ultimate result.