In this paper, we introduce the concept of $$\alpha $$ -fractal function and fractal approximation for a set-valued continuous map defined on closed bounded interval real numbers. Also, study some properties such functions. Further, estimate perturbation error between given its function. Additionally, define new graph different from standard introduced in literature establish bounds dimension n...

The article is a continuation of [1]. We define the following concepts: a function from a set X into a set Y , denoted by “Function of X ,Y ”, the set of all functions from a set X into a set Y , denoted by Funcs(X ,Y ), and the permutation of a set (mode Permutation of X , where X is a set). Theorems and schemes included in the article are reformulations of the theorems of [1] in the new termi...