Volterra's function: Difference between revisions

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{{Short description|Differentiable function whose derivative is not Riemann integrable}}

{{more citations needed|date=August 2024}}

[[File:VolerraVolterra function.svg|thumb|400px|right|The first three steps in the construction of Volterra's function.]]

In [[mathematics]], '''Volterra's function''', named for [[Vito Volterra]], is a [[real-valued function]] ''V'' [[Function of a real variable|defined on the real line]] '''R''' with the following curious combination of properties:

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* ''V'' is [[Differentiable function|differentiable]] everywhere

* The derivative ''V'' ′ is [[bounded function|bounded]] everywhere

* The derivative is not [[Riemann integration|Riemann-integrable]].<ref>{{Cite web |title=Ouvrages de référence — Wikipédia |url=https://fr.wikipedia.org/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/0-8218-3805-9 |access-date=2024-08-12 |website=fr.wikipedia.org |language=fr}}</ref>

==Definition and construction==

The function is defined by making use of the [[Smith–Volterra–Cantor set]] and an infinite number or "copies" of sections of the function defined by

The function is defined by making use of the [[Smith–Volterra–Cantor set]] and "copies" of the function defined by <math>f(x) = x^2 \sin(1/x)</math> for <math>x \neq 0</math> and <math>f(0) = 0</math>. The construction of ''V'' begins by determining the largest value of ''x'' in the interval [0, 1/8] for which ''f'' &prime;(''x'') = 0. Once this value (say ''x''₀) is determined, extend the function to the right with a constant value of ''f''(''x''₀) up to and including the point 1/8. Once this is done, a mirror image of the function can be created starting at the point 1/4 and extending downward towards 0. This function will be defined to be 0 outside of the interval [0, 1/4]. We then translate this function to the interval [3/8, 5/8] so that the resulting function, which we call ''f''₁, is nonzero only on the middle interval of the complement of the Smith–Volterra–Cantor set. To construct ''f''₂, ''f'' &prime; is then considered on the smaller interval [0,1/32], truncated at the last place the derivative is zero, extended, and mirrored the same way as before, and two translated copies of the resulting function are added to ''f''₁ to produce the function ''f''₂. Volterra's function then results by repeating this procedure for every interval removed in the construction of the Smith–Volterra–Cantor set; in other words, the function ''V'' is the limit of the sequence of functions ''f''₁, ''f''₂, ...

:<math>f(x) = \begin{cases} x^2 \sin(1/x), & x \ne 0 \\ 0, & x = 0.\end{cases}</math>

The function is defined by making use of the [[Smith–Volterra–Cantor set]] and "copies" of the function defined by <math>f(x) = x^2 \sin(1/x)</math> for <math>x \neq 0</math> and <math>f(0) = 0</math>. The construction of ''V'' begins by determining the largest value of ''x'' in the interval [0, 1/8] for which ''f'' &prime;(''x'') = 0. Once this value (say ''x''₀) is determined, extend the function to the right with a constant value of ''f''(''x''₀) up to and including the point 1/8. Once this is done, a mirror image of the function can be created starting at the point 1/4 and extending downward towards 0. This function will be defined to be 0 outside of the interval [0, 1/4]. We then translate this function to the interval [3/8, 5/8] so that the resulting function, which we call ''f''₁, is nonzero only on the middle interval of the complement of the Smith–Volterra–Cantor set. To construct ''f''₂, ''f'' &prime; is then considered on the smaller interval [0,1/32], truncated at the last place the derivative is zero, extended, and mirrored the same way as before, and two translated copies of the resulting function are added to ''f''₁ to produce the function ''f''₂. Volterra's function then results by repeating this procedure for every interval removed in the construction of the Smith–Volterra–Cantor set; in other words, the function ''V'' is the limit of the sequence of functions ''f''₁, ''f''₂, ...

To construct ''f''₂, ''f'' &prime; is then considered on the smaller interval [0,1/32], truncated at the last place the derivative is zero, extended, and mirrored the same way as before, and two translated copies of the resulting function are added to ''f''₁ to produce the function ''f''₂. Volterra's function then results by repeating this procedure for every interval removed in the construction of the Smith–Volterra–Cantor set; in other words, the function ''V'' is the limit of the sequence of functions ''f''₁, ''f''₂, ...

==Further properties==