Approximation by the linear fractal interpolation functions with the same fractal dimension

A continuous function defined on a closed interval can be of unbounded variation with certain fractal dimension. Fractal interpolation functions are often used to approximate such functions whose structure seem self affine. In the present paper, a continuous function with non-integer Box dimension has been approximated by certain linear fractal interpolation functions. Both the original function and the linear fractal interpolation functions have the same Box dimension. The interpolation approximation of integer dimensional continuous functions has also been discussed elementary.