On Fractal Features and Fractal Linear Space About Fractal Continuous Functions

Abstract: This paper investigates fractal dimension of linear combination of fractal continuous functions with the same or different fractal dimensions. It has been proved that: (1) $BV_{I}$ all fractal continuous functions with bounded variation is fractal linear space; (2) ${}^{1}D_{I}$ all fractal continuous functions with Box dimension one is a fractal linear space; (3) ${}^{s}D_{I}$ all fractal continuous functions with identical Box dimension $s(1<s\leq 2)$ is surprisingly a non-fractal linear space, even non-fractal linear manifold, beyond our initial expectation, because the Box dimension of linear combination of fractal continuous functions can take any real number in $[1,s)$ if it exists, and some different upper and lower Box dimension if it does not exit. This attracts our interests to probe into fractal characteristics of ${}^{s}D_{I}$, and get some suggesting results.