Riemann's Example of a Continuous but Nowhere Differentiable Function

2007 ◽  
Keyword(s):  
Differentiable Function ◽  
Nowhere Differentiable Function

Mathematics

2019 ◽  
pp. 502-543
Author(s):  
Paul Rusnock ◽  
Jan Šebestík
Keyword(s):  
Mathematical Method ◽  
Differentiable Function ◽  
Infinite Series ◽  
Historical Context ◽  
Real Analysis ◽  
Analytic Proof ◽  
Nowhere Differentiable Function

This chapter presents an overview of Bolzano’s work in mathematics and its philosophy, while presenting some interesting samples of his work. It begins with a discussion of his views on mathematical method in their historical context, followed by an exposition of some of his best work in real analysis. In particular, the chapter discusses his early work on infinite series and his analysis of continuity, beginning with the Purely Analytic Proof (1817), and extending to his construction of a continuous, nowhere differentiable function in the 1830s, called Bolzano’s function. (85 words)

CONVERGENCE OF THE WEIERSTRASS-MANDELBROT PROCESS TO FRACTIONAL BROWNIAN MOTION

Fractals ◽  
2000 ◽  
Vol 08 (04) ◽  
pp. 369-384 ◽  
Author(s):  
VLADAS PIPIRAS ◽  
MURAD S. TAQQU
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Differentiable Function ◽  
General Type ◽  
Nowhere Differentiable Function ◽  
Dependence Structures

In this work, we develop Mandelbrot's idea that Weierstrass's nowhere differentiable function can be modified and randomized to approximate fractional Brownian motion (FBM). Our approach covers the convergence of processes of a more general type and allows us to consider different dependence structures in the above randomization.

WEIERSTRASS-TYPE FUNCTIONS IN p-ADIC LOCAL FIELDS

Fractals ◽  
2020 ◽  
Vol 28 (03) ◽  
pp. 2050043
Author(s):  
YIN LI
Keyword(s):  
Dynamical System ◽  
Differentiable Function ◽  
Local Field ◽  
Discrete Dynamical System ◽  
Linear Connection ◽  
Local Fields ◽  
Type Function ◽  
Nowhere Differentiable Function

The Weierstrass nowhere differentiable function has been studied often as example of functions whose graphs are fractals in [Formula: see text]. This paper investigates the Weierstrass-type function in the [Formula: see text]-adic local field [Formula: see text] whose graph is a repelling set of a discrete dynamical system, and proves that there exists a linear connection between the orders of the [Formula: see text]-adic calculus and the dimensions of the corresponding graphs.

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