Funky Functions
Math 555
Prof. Girardi

To find 7 funky functions:
Below are further comments on these 7 functions.

Seven Funky Functions

1. Dirichlet function
• The Dirichlet Function is not continuous at any point on the real line.
• A variant of Trench's Exercise 2.2.6.
• Wiki page
2. Countable discontinuities
• A function that is continuous at the irrational numbers and discontinuous at the rational numbers.
• Has many names, such as: Thomae's function, popcorn function, raindrop function.
• A variant of Trench's Exercise 2.2.7.
• Wiki page
3. C1 function
4. Cn function
5. Cinf function
6. Weierstrass function
• A function that is continuous everywhere and nowhere differentiable in R.
• A Weierstrass Function is a function of the form
Σ n=0 &infin   an cos (bn π x)           or           Σ n=0 &infin   an sin (bn π x)
where there are some kind of restriction on a and b.
• A Weierstrass Function is everywhere continuous but nowhere differentiable if 0 < a < 1 and ab ≥ 1 , see (Hardy G.H., Weierstrass's nondifferentiable function, Trans - Amer. Math. Soc, 17(1916), 301-325).
• From the Weierstrass Function's Wiki page:  It turns out that the Weierstrass function is far from being an isolated example: although it is "pathological", it is also "typical" of continuous functions. In a topological sense, it can be shown that the set of nowhere-differentiable real-valued functions on [0, 1] is dense in the vector space C([0, 1]; R) of all continuous real-valued functions on [0, 1] with the topology of uniform convergence.
• Katie Spurrier's SCHC Senior Thesis Continuous Nowhere Differentiable Functions
• Excuted Maple Worksheet in HTML
• Maple Woksheet in Maple
7. Cantor function

The Maple Worksheets above prepared by Prof. Girardi. Unless otherwise stated, Maple Code in these Maple Worksheets were prepared by Prof. Girardi.

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