Overview of One-Dimensional Continuous Functions with Fractional Integral and Applications in Reinforcement Learning

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 hodlder 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.

Automated summary

One-dimensional continuous functions are fundamental in studying complex functions. The properties of these functions, such as dimensionality, continuity, and boundedness, have been discussed from various perspectives. This article systematically sorts out existing conclusions based on bounded variation, unbounded variation, and Hodler continuity. Additionally, it explores the innovative use of unbounded variation points in analyzing continuous functions and constructing functions with unbounded variation. The potential applications of fractals and fractal dimension in reinforcement learning are also predicted.