To find 7 funky functions:
Below are further comments on these 7 functions.
Seven Funky Functions
The Dirichlet Function is not continuous at any point on the real line.
A variant of Trench's Exercise 2.2.6.
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.
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)
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
0 < a < 1 and ab ≥ 1 ,
(Hardy G.H., Weierstrass's nondifferentiable function, Trans - Amer. Math. Soc, 17(1916), 301-325).
- From the Weierstrass Function's
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
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