  In this paper, the notion of dimension preserving approximation for
real-valued bivariate continuous functions, defined on a rectangular domain
$\rectangle$, has been introduced and several results, similar to well-known
results of bivariate constrained approximation in terms of dimension preserving
approximants, have been established. Further, some clue for the construction of
bivariate dimension preserving approximants, using the concept of fractal
interpolation functions, has been added. In the last part, some multi-valued
fractal operators associated with bivariate $\alpha$-fractal functions are
defined and studied.
