Since fractal functions are widely applied in dynamic systems and physics such as fractal growth and fractal antennas, this paper concerns fundamental problems of fractal continuous functions like cardinality of collection of fractal functions, box dimension of summation of fractal functions, and fractal linear space. After verifying that the cardinality of fractal continuous functions is the second category by Baire theory, we investigate the box dimension of sum of fractal continuous functions so as to discuss fractal linear space under fractal dimension. It is proved that the collection of 1-dimensional fractal continuous functions is a fractal linear space under usual addition and scale multiplication of functions. Particularly, it is revealed that the fractal function with the largest box dimension in the summation represents a fractal dimensional character whenever the other box dimension of functions exist or not. Simply speaking, the fractal function with the largest box dimension can absorb the other fractal features of functions in the summation.

由于分形函数广泛应用于分形生长、分形天线等动态系统和物理领域，本文主要研究分形连续函数的基本问题，如分形函数集合的基数、分形函数求和的盒维数、分形线性空间等。在用Baire理论验证了分形连续函数的基数是第二类后，我们研究了分形连续函数和的盒维数，以讨论分形维数下的分形线性空间。证明了一维分形连续函数的集合在函数的通常加法和尺度乘法下是一个分形线性空间。特别，揭示了求和中具有最大盒维数的分形函数代表了一个分形维数，只要函数的其他盒维数存在与否。简单来说，具有最大盒子维数的分形函数可以吸收求和中函数的其他分形特征。