Simple proofs of the continuity and nowhere differentiability of fractal curves
(2021) In Bachelor's Theses in Mathematical Sciences MATK11 20211Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
- Abstract
- Fractal curves are self similar objects, most of which have the property of being nowhere differentiable. The foundations of fractal curves are in the study of continuous functions with no points of differentiability. In this thesis, the continuity and nowhere differentiability of the Weierstrass function will be proved by using an auxiliary function and mostly elementary tools from Fourier analysis and integration theory. Furthermore, by identifying some of the properties of the Weierstrass function, the method of the proof will be generalised to set up and prove a theorem that can be used to determine the existence of continuous functions that have no points of differentiability. Finally, a modified version of that theorem will be set up... (More)
- Fractal curves are self similar objects, most of which have the property of being nowhere differentiable. The foundations of fractal curves are in the study of continuous functions with no points of differentiability. In this thesis, the continuity and nowhere differentiability of the Weierstrass function will be proved by using an auxiliary function and mostly elementary tools from Fourier analysis and integration theory. Furthermore, by identifying some of the properties of the Weierstrass function, the method of the proof will be generalised to set up and prove a theorem that can be used to determine the existence of continuous functions that have no points of differentiability. Finally, a modified version of that theorem will be set up and proved, where some of the conditions in the theorem will be weakened so that the theorem may be used to identify a wider range of continuous but nowhere differentiable functions. (Less)
- Popular Abstract
- One of the most fundamental concepts in mathematics is that of the derivative, a function that describes rate of change. When the derivative of a function exists at every point in its domain, that function is called differentiable. For a function to be differentiable, it must also be continuous, meaning there are no breaks or holes in the graph of the function. Contrary to popular belief that lasted until the late 19th Century, the converse is not true; continuous functions need not be differentiable and in fact, most continuous functions are not. The most well-known example of a continuous function with no points of differentiability is the Weierstrass function, presented by the German mathematician Karl Weierstrass in 1872. The... (More)
- One of the most fundamental concepts in mathematics is that of the derivative, a function that describes rate of change. When the derivative of a function exists at every point in its domain, that function is called differentiable. For a function to be differentiable, it must also be continuous, meaning there are no breaks or holes in the graph of the function. Contrary to popular belief that lasted until the late 19th Century, the converse is not true; continuous functions need not be differentiable and in fact, most continuous functions are not. The most well-known example of a continuous function with no points of differentiability is the Weierstrass function, presented by the German mathematician Karl Weierstrass in 1872. The Weierstrass function is an infinite sum of cosines, each scaled by some constant; this type of sum is known as a Fourier series. The Weierstrass function also has the property of self-similarity, which makes it an example of a fractal curve. There have been several proofs of the continuity and nowhere differentiability of the Weierstrass function. This thesis will follow a recent approach that uses elementary tools from Fourier analysis and integration theory to prove that the Weierstrass function is continuous and nowhere differentiable. The proof will be generalised to prove two theorems, one more general than the other, which can be used to determine the existence of continuous but nowhere differentiable functions. Functions that are continuous but nowhere differentiable have been found to have many useful applications. For instance, they can be used to describe random behaviour that evolves over time, a concept known as Brownian motion which has popular applications in physics and economics among other disciplines. Fractals tend to occur as natural phenomena, which makes them widely applicable. For example, eroded coastlines and the blood vessels are such structures that can be modelled by fractal curves. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9057844
- author
- Felician Kouwenhoven, Eva Florence LU
- supervisor
-
- Nils Dencker LU
- organization
- course
- MATK11 20211
- year
- 2021
- type
- M2 - Bachelor Degree
- subject
- keywords
- Weierstrass function, Nowhere-differentiability, Fractal, Fourier Analysis
- publication/series
- Bachelor's Theses in Mathematical Sciences
- report number
- LUNFMA-4116-2021
- ISSN
- 1654-6229
- other publication id
- 2021:K19
- language
- English
- id
- 9057844
- date added to LUP
- 2022-08-08 17:39:01
- date last changed
- 2022-08-08 17:39:01
@misc{9057844,
abstract = {{Fractal curves are self similar objects, most of which have the property of being nowhere differentiable. The foundations of fractal curves are in the study of continuous functions with no points of differentiability. In this thesis, the continuity and nowhere differentiability of the Weierstrass function will be proved by using an auxiliary function and mostly elementary tools from Fourier analysis and integration theory. Furthermore, by identifying some of the properties of the Weierstrass function, the method of the proof will be generalised to set up and prove a theorem that can be used to determine the existence of continuous functions that have no points of differentiability. Finally, a modified version of that theorem will be set up and proved, where some of the conditions in the theorem will be weakened so that the theorem may be used to identify a wider range of continuous but nowhere differentiable functions.}},
author = {{Felician Kouwenhoven, Eva Florence}},
issn = {{1654-6229}},
language = {{eng}},
note = {{Student Paper}},
series = {{Bachelor's Theses in Mathematical Sciences}},
title = {{Simple proofs of the continuity and nowhere differentiability of fractal curves}},
year = {{2021}},
}