The classification of functions of unbounded variation on closed intervals

Abstract

The present paper mainly discuss the classification of continuous and discontinuous functions of unbounded variation on closed intervals. The classification of functions of bounded variation on closed intervals has been discussed. Continuous functions of unbounded variation on closed intervals have been also proved to having at least one unbounded variation points, which are also called as unbounded variation functions, singular fractal functions, irregular fractal functions and regular fractal functions constitute fractal functions. Discontinuous fractal functions have also been investigated. Corresponding to condition of continuous functions, discontinuous singular fractal functions, discontinuous irregular fractal functions and discontinuous regular fractal functions can also be explored by length of graphs of those functions instead of variation. Continuous functions can also be classified as differentiable functions and nondifferentiable functions. All differentiable functions are of bounded variation Nondifferentiable functions are composed of bounded variation functions and unbounded variation functions. Fractal dimension of all bounded variation continuous functions is 1. Furthermore, unbounded variation continuous functions may have finite unbounded variation points or infinite unbounded variation points. Number of unbounded variation points of unbounded variation continuous functions maybe infinite and countable or uncountable. Certain examples of different unbounded variation continuous functions have been given in this paper.