Bernd Kirchheim, Paul F. X. Müller · Proceedings of the American Mathematical Society · 2008
A real-valued continuously differentiable function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the unit interval is constructed such that <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript k equals 1 Overscript normal infinity Endscripts beta Subscript f Baseline left-parenthesis x comma 2 Superscript negative k Baseline right-parenthesis equals normal infinity"> <mml:semantics> <mml:mrow> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:munderover> <mml:msub> <mml:mi> β </mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sum _{k=1}^\infty \beta _f (x, 2^{-k} ) = \infty</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> holds for every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of left-bracket 0 comma 1 right-bracket period"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">x \in [0,1].</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Here <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta Subscript f Baseline left-parenthesis x comma 2 Superscript negative k Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> β </mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta _f (x, 2^{-k} )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> measures the distance of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the best approximating linear function at scale <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript negative k"> <mml:semantics> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">2^{-k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> around <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .
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Müller, P. F. X., Kirchheim, B. · Max Planck Institute for Plasma Physics · 2008
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T. W. Körner, Terence Tao · Cambridge University Press eBooks · 2022
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G. H. Hardy · Transactions of the American Mathematical Society · 1916
Cited by 127
G. H. Hardy · Transactions of the American Mathematical Society · 1916
1. Introduction. 301 2. Weierstrass's function when 6 is an integer. 304 3. Weierstrass's function when b is not an integer. 314 4. Other functions. 320 4.1. A function which does not satisfy a Lipschitz condition of any order. 320 4.2. On a theorem of S. Bernstein. 321 4.3. Riemann's non-differentiable function. 322
Cited by 311Open access
Shyam Kamal, Xinghuo Yu, Rahul Kumar Sharma et al. · IEEE Transactions on Circuits & Systems II Express Briefs · 2019
The tracking problem of desired signal can be transformed to an equivalent stabilization problem about the origin in the error co-ordinate provided the desired signal is differentiable. If the desired signal is not differentiable, then it is not straightforward to solve the tracking problem. In this brief, a methodology based on fractional operators has been suggested to solve this tracking problem, for those classes of non-differentiable reference signals which are integrable. The simulation results of its application to a switch controlled RL circuit is demonstrated.
Cited by 8
Feng Jiang · 2010
There is a close connection between Riemann differentiable function and continuous function.The essay tries to prove that Riemann differentiable function on closed interval [a,b] is continuous function on density set of [a,b],and one related example is given as well.
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Raymond O. Wells · Elsevier eBooks · 1974
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Nishan Krikorian · Transactions of the American Mathematical Society · 1972
A <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript s"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^s}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> differentiable manifold structure is constructed for spaces of maps from a compact <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript r"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^r}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript r plus s"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> <mml:mo>+</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^{r + s}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The method (1) is inspired by Douady; (2) does not require any additional structure on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (such as sprays); (3) includes the case when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an analytic manifold and concludes that the mapping space is also an analytic manifold; (4) can be used to treat all the classical mapping spaces (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript r"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^r}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> functions, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript r"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^r}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> functions with Hölder conditions, and Sobolev functions). Several interesting aspects of these manifolds are investigated such as their tangent spaces, their behavior with respect to functions, and realizations of Lie group structures on them. Differentiable structures are also exhibited for spaces of compact maps with noncompact domain.
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Mark Lynch · American Mathematical Monthly · 1992
(1992). A Continuous, Nowhere Differentiable Function. The American Mathematical Monthly: Vol. 99, No. 1, pp. 8-9.
Cited by 17
Wen Liu · American Mathematical Monthly · 2000
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Delio Vicini, Sébastien Speierer, Wenzel Jakob · ACM Transactions on Graphics · 2022
Physically-based differentiable rendering has recently emerged as an attractive new technique for solving inverse problems that recover complete 3D scene representations from images. The inversion of shape parameters is of particular interest but also poses severe challenges: shapes are intertwined with visibility, whose discontinuous nature introduces severe bias in computed derivatives unless costly precautions are taken. Shape representations like triangle meshes suffer from additional difficulties, since the continuous optimization of mesh parameters cannot introduce topological changes. One common solution to these difficulties entails representing shapes using signed distance functions (SDFs) and gradually adapting their zero level set during optimization. Previous differentiable rendering of SDFs did not fully account for visibility gradients and required the use of mask or silhouette supervision, or discretization into a triangle mesh. In this article, we show how to extend the commonly used sphere tracing algorithm so that it additionally outputs a reparameterization that provides the means to compute accurate shape parameter derivatives. At a high level, this resembles techniques for differentiable mesh rendering, though we show that the SDF representation admits a particularly efficient reparameterization that outperforms prior work. Our experiments demonstrate the reconstruction of (synthetic) objects without complex regularization or priors, using only a per-pixel RGB loss.
Cited by 105Open access
Nishan Krikorian · Transactions of the American Mathematical Society · 1972
A ${C^s}$ differentiable manifold structure is constructed for spaces of maps from a compact ${C^r}$ manifold $M$ to a ${C^{r + s}}$ manifold $N$. The method (1) is inspired by Douady; (2) does not require any additional structure on $N$ (such as sprays); (3) includes the case when $N$ is an analytic manifold and concludes that the mapping space is also an analytic manifold; (4) can be used to treat all the classical mapping spaces (${C^r}$ functions, ${C^r}$ functions with Hölder conditions, and Sobolev functions). Several interesting aspects of these manifolds are investigated such as their tangent spaces, their behavior with respect to functions, and realizations of Lie group structures on them. Differentiable structures are also exhibited for spaces of compact maps with noncompact domain.
Cited by 1
Vladimir Averbuch, Томас Кондерла · Mathematical Notes · 2012
We construct here an example of a continuous convex function on a locally convex space, which is MS-differentiable at a point, but is not HL-differentiable at this point.
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T. W. Körner · Colloquium Mathematicum · 2020
The convolution product of a differentiable function with itself may fail to be Hölder continuous of any index on a dense set of points.
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V. F. Demyanov, Lyudmila N. Polyakova · USSR Computational Mathematics and Mathematical Physics · 1980
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Sunghyu Han, O. L. Mangasarian · 1981
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Hafner, Izidor · Acervo Digital da Universidade Estadual Paulista (Universidade Estadual Paulista) · 2011
Bolzano discovered this continuous but nowhere differentiable function before 1831, but these investigations were not published until 1930. Weierstrass found an analogous function in 1875. The function is the limit of the ones graphed as n goes to infinity
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Sunghyu Han, O. L. Mangasarian · Mathematical Programming · 1983
Cited by 33
John E. McCarthy · American Mathematical Monthly · 1953
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