  In this article, we construct the multivariate fractal interpolation
functions for a given data points and explore the existence of $\alpha$-fractal
function corresponding to the multivariate continuous function defined on
$[0,1]\times \cdots \times [0,1](q\text{-times})$. The parameters are selected
such that the corresponding fractal version preserves some of the original
function's properties, for instance, if the given function is H\"older
continuous, then the corresponding $\alpha$-fractal function is also H\"older
continuous. Moreover, we explore the restriction of the $\alpha$-fractal
function on the co-ordinate axis. Furthermore, the box dimension and Hausdorff
dimension of the graph of the multivariate $\alpha$-fractal function and its
restriction are investigated. In the last section, we prove that the mixed
Riemann-Liouville fractional integral of fractal function satisfies a
self-referential equation.
