Simple Proofs of Nowhere-Differentiability for Weierstrass’s Function and Cases of Slow Growth

Johnsen, Jon

doi:10.1007/s00041-009-9072-2

Cited by 28 publications

(8 citation statements)

References 14 publications

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“…Multiple proofs of the Weierstrass function's non-differentiability at all points have been published; several of these use Fourier Series, which goes beyond the space constraints of this paper (and the time constraints of its author). For a complete proof of Theorem 2, reference [6,7].…”

Section: Proof Of Uniform Continuity and Nowhere Differentiabilitymentioning

confidence: 99%

On French Pudding and a German Mathematician

Shoemaker¹

2017

jhummath

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At the turn of the 19 th century, mathematics developed the rigor that is now considered essential to the field. Mathematicians began going back and proving theorems and statements that had been taken to be true on face value, ensuring the underpinnings of mathematics were solid. This era of mathematics was characterized not only by setting foundations, but also by pushing the boundaries of new ideas. In 1830, Bolzano found an example of a function that was nowhere differentiable, despite being continuous. Thirty years later, Cellerier and Riemann each discovered another example of such a pathological function. The first everywhere continuous nowhere differentiable function to be published appeared in Borchardt's Journal in 1875. This function was proposed by Karl Weierstrass. His function, in addition to similar examples that followed it, revolutionized the ideas of continuity, differentiability, and limits. Due to these pathological functions, the traditional definitions were rethought and revised. Here we explore some of these pathological functions.

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Section: Proof Of Uniform Continuity and Nowhere Differentiabilitymentioning

confidence: 99%

On French Pudding and a German Mathematician

Shoemaker¹

2017

jhummath

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“…Many authors found the fractional derivative of the continuous but nowhere differentiable function that is Weierstrass Function [10][11][12][13][14][15][16][17] using different type definitions of fractional derivatives. Here Jumarie type fractional order derivative of ( ) W x α α is of order α…”

Section: The Jumarie Fractional Derivative Of Fractional Weierstrass mentioning

confidence: 99%

“…On the other hand fractional calculus is another developing mathematical tool to study the continuous but non-differentiable functions (signals) where the conventional calculus fails [4][5][6][7][8][9][10][11]. Many authors are trying to relate between the fractional derivative and fractional dimension [1,[12][13][14][15]. The functions which are continuous but non-differentiable in integer order calculus can be characterized in terms of fractional calculus and especially through Holder exponent [10,16].…”

Section: Introductionmentioning

confidence: 99%

“…The functions which are continuous but non-differentiable in integer order calculus can be characterized in terms of fractional calculus and especially through Holder exponent [10,16]. To study the no-where differentiable functions authors in [12][13][14][15][16] used different type of fractional derivatives. Jumarie [17] defines the fractional trigonometric functions in terms of Mittag-Leffler function and established different useful fractional trigonometric formulas.…”

Section: Introductionmentioning

confidence: 99%

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Fractional Weierstrass Function by Application of Jumarie Fractional Trigonometric Functions and Its Analysis

Ghosh¹,

Sarkar²,

Das

2015

APM

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The classical example of no-where differentiable but everywhere continuous function is Weierstrass function. In this paper we define the fractional order Weierstrass function in terms of Jumarie fractional trigonometric functions. The Holder exponent and Box dimension of this function are calculated here. It is established that the Holder exponent and Box dimension of this fractional order Weierstrass function are the same as in the original Weierstrass function, independent of incorporating the fractional trigonometric function. This is new development in generalizing the classical Weierstrass function by usage of fractional trigonometric function and obtain its character and also of fractional derivative of fractional Weierstrass function by Jumarie fractional derivative, and establishing that roughness index are invariant to this generalization.

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“…In this paper, we study the uniform Hölder continuity of R α,β with β ≥ α − 1 in order to complete and generalize a result of Johnsen [19] in 2010 which claims that, if β > α−1, then R α,β is uniformly Hölder continuous with an exponent superior or equal to (α − 1)/β. To achieve this, we use some techniques different from the ones of Johnsen.…”

Section: Introductionmentioning

confidence: 99%