The Fractal Properties Of The Nowhere Differentiable Continuous Function

Continuity and differentiability are important components of classical analysis. The appearance of weierstrass function lays a foundation of the continuity and nowhere differentiable function, besides pioneers a new research area. With the establishment of a new mathematical subject fractal more and more people dedicate to such work .On 1870s',some mathematicians combine the continuous and nowhere differentiable function with fractal and pioneer an important research area again. The paper mainly considers some fractal dimensions of the continuity and differentiability function and then gets some results.Firstly, the appearance, the development and the main theory of the continuity and nowhere differentiable function are introduced. Secondly several construction methods of the continuity and nowhere differentiable function are investigated and the theory about whether the function is nowhere differentiable and continuous is proved, too. At the same time, some properties, fractal dimensions and Holder continuity property about nowhere differentiable function are also proposed. Finally, we focus on the fractal properties of some nowhere differentiable continuous functions, including some conclusion about its concept, estimate of fractal dimension and Holder continuous property.The research on the fractal property of nowhere differentiable continuous functions is still under furthermore investigation and development. The aim is to calculate its fractal dimension, so give some theory basis for practical application of fractal.