Based on the previous studies, we make further research on how fractal dimensions of graphs of fractal continuous functions under operations change and obtain a series of new results in this paper. Initially, it has been proven that a positive continuous function under unary operations of any nonzero real power and the logarithm taking any positive real number that is not equal to one as the base number can keep the fractal dimension invariable. Then, a general method to calculate the Box dimension of two continuous functions under binary operations has been proposed. Using this method, the lower and upper Box dimensions of the product and the quotient of continuous functions without zero points have been investigated. On this basis, these conclusions will be generalized to the ring of rational functions. Furthermore, we discuss the Hölder continuity of continuous functions under operations and then prove that a Lipschitz function can be absorbed by any other continuous functions under certain binary operations in the sense of fractal dimensions. Some elementary results for vector-valued continuous functions have also been given.

本文在前人研究的基础上，对分形连续函数图的分形维数在运算条件下如何变化进行了进一步研究，并取得了一系列新的成果。初步证明，任意非零实次幂和以任意不等于1的正实数为基数的对数一元运算下的正连续函数可以保持分形维数不变。然后，提出了一种计算二元运算下两个连续函数的Box维数的通用方法。利用该方法，研究了乘积的上下 Box 维数以及无零点的连续函数的商。在此基础上，将这些结论推广到有理函数环上。此外，我们讨论了连续函数在运算下的Hölder连续性，然后证明了在分形维数意义上，在某些二元运算下，Lipschitz函数可以被任何其他连续函数吸收。还给出了向量值连续函数的一些基本结果。