Abstract
One-dimensional continuous functions are important fundament for studying other complex functions. Many theories and methods applied to study one-dimensional continuous functions can also be accustomed to investigating the properties of multi-dimensional functions. The properties of one-dimensional continuous functions, such as dimensionality, continuity, and boundedness, have been discussed from multiple perspectives. Therefore, the existing conclusions will be systematically sorted out according to the bounded variation, unbounded variation and ho¨lder continuity. At the same time, unbounded variation points are used to analyze continuous functions and construct unbounded variation functions innovatively. Possible applications of fractal and fractal dimension in reinforcement learning are predicted.
Highlights
There is a growing body of literature that recognises the importance of using fractal dimension instead of topological dimension to describe the functions
The unbounded variation function can be defined by the complementary set of bounded variation functions, but this paper will research unbounded variation functions through the unbounded variation point that can be found in Definition 3
There has been a lot of research on using fractals to improve search efficiency [54,55,56], but these algorithms can still continue to be optimized. This manuscript systematically sorts out the conclusions about one-dimensional continuous functions
Summary
It is a widely held view that dimensionality is an important indicator to describe functions, but different functions have many disparate internal structures and properties. The Box dimension of bounded variation functions and the functions with Riemann-Liuville fractional calculus are both one. The Box dimension of an unbounded variation function with only an unbounded variation point is one If this function has self-similarity at the same time, its Hausdorff dimension is one. Scholars are not very familiar with the image of any one-dimensional continuous functions with an unbounded variation point.
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