Andrew HwangF PeriodDeggeller

Continuous everywhere but differentiable nowhere functions and fractals

Our group’s final project branched off of the topic of the Weierstrass

function, a function that is continuous everywhere yet differentiable nowhere. The

Weierstrass function is shown above. We decided to go further and explore other

functions that had this same property, one being the blancmange function.

Branching off even further, our group explored fractals, as the Weierstrass function

and blancmange function are technically fractals.

In this project, I cover the blancmange function, otherwise known as the

Takagi fractal curve. I also did fractals in general, especially the properties of

fractals. Along with the properties, I also covered the general definition of fractals

and finding pictures of fractals.

In doing research for the Weierstrass function, I found that there were more

continuous everywhere yet differentiable

nowhere functions. The most famous among

these is the blancmange function, also known

as the Takagi-Landsberg curve. To construct

this curve, one can take a sawtooth function,

much like a negative absolute value function, and continually increasing the

periodicity of the function while halving the height of the function and infinite

number of times. All iterations are added together, creating a graph like the one

shown to the right. The blancmange function’s equation is blanc( x )=∑

n=0

¥ s (2n x )2n .

The function “s” is the distance from x to the closest integer. The Takagi-Landsberg

curve is a generalization of the blancmange function, given by the equation

T w (x )=∑n=0

¥

wn s(2n x ). In this equation, “w” is a number; if “w” is ½, then the

blancmange function is given. Interestingly, if “w” is ¼, the function yields a

parabola.

As a connection to the next part of our project, I used the blancmange and

Weierstrass functions to transition into fractals, geometric shapes that can be split

into parts, each being a reduced copy of the

entire graph. In identifying fractals, certain

properties must be observed. Among these

include self-similarity, or the repetition of

patterns at all scales, and infinite complexity and detail. Fractals must also be too

irregular to be described in terms of geometry; therefore, a line is not considered a

fractal. Other properties include being able to enclose a finite area with one single

line, and being formed through iterative process, where some process that makes

the shape is repeated infinitely.