# Continuous Nowhere Differentiable Functions

@inproceedings{Thim2003ContinuousND, title={Continuous Nowhere Differentiable Functions}, author={Johan Thim}, year={2003} }

In the early nineteenth century, most mathematicians believed that a continuous function has derivative at a significant set of points. A.~M.~Amp\`ere even tried to give a theoretical justificati ...

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#### References

SHOWING 1-10 OF 120 REFERENCES

The prevalence of continuous nowhere differentiable functions

- Mathematics
- 1994

In the space of continuous functions of a real variable, the set of nowhere differentiable functions has long been known to be topologically "generic". In this paper it is shown further that in a… Expand

A Nowhere Differentiable Continuous Function

- Mathematics, Computer Science
- Am. Math. Mon.
- 2000

The purpose of this note is to give a new elementary example that needs only the basic notion of limit and is very accessible and appropriate for a first calculus course. Expand

A continuous, nowhere differentiable function

- Mathematics
- 1992

The examples of continuous, nowhere differentiable functions given in most analysis and topology texts involve the uniform limit of a series of functions in the former and the Baire category theorem… Expand

A Nowhere Differentiable Continuous Function Constructed Using Cantor Series

- Mathematics
- 2001

The examples of continuous nowhere differentiable functions given in most analysis texts involve the uniform convergence of a series of functions (see Hobson [1, pp. 401412]). In the last twenty… Expand

The set of continuous nowhere differentiable functions

- Mathematics
- 1979

Let C be the space of all real-valued continuous functions defined on the unit interval provided with the uniform norm. In the Scottish Book, Banach raised the question of the descriptive class of… Expand

Continuous nowhere-differentiable functions—an application of contraction mappings

- Mathematics
- 1991

then f is also a continuous nowhere-differentiable function. (See [3, p. 115].) The above examples have concise definitions and establish the existence of continuous nowhere-differentiable functions.… Expand

Prevalence: an addendum

- Mathematics
- 1993

The authors mention work of Christensen, Mycielski, Tsujii, and others which is closely related to a survey article by the first author [math.FA/9210220].

An Elementary Proof that Schoenberg's Space-Filling Curve Is Nowhere Differentiable

- Mathematics
- 1992

in which the functions f, g are continuous on the interval 1", or by words to that effect. This "conventional wisdom" was shattered in 1890 when G. Peano [1] demonstrated that this definition… Expand

A Unified Approach for Nondifferentiable Functions

- Mathematics
- 1994

Abstract Let ψ( x ) denote the distance between x and the nearest integer, and fix 0 a ab > 1, where b is not necessarily an integer. Then for any sequence θ n of phases, the function f ( x ) = ∑ ∞ n… Expand

On the Continuity of van der Waerden's Function in the Hölder Sense

- Mathematics
- 1986

It is well known that the number v is transcendental; the proof is complicated. However, there are several elementary proofs that X and v 2 are irrational; see for example [1]. Using a simple trick,… Expand