  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.
