In this paper we present a method for constructing the continuous best fractal approximation in the space of bounded functions. We construct the finite-dimensional subspace of the space of bounded functions whose base consists of the continuous fractal functions, and propose how to find the best approximation of given continuous function by element of the constructed space.

We present a method constructing a function which is the best approximation for given data and satisfiesthe given self-similar condition. For this, we construct a space F of local self-similar fractal functions and show its properties. Next we present a computational scheme constructing the best fractal approximation in this space and estimate an error of the constructed fractal approximation. Our best fractal approximation is a fixed point of some fractal interpolation function.