Volterra's Function
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Volterra's function, named for
Vito Volterra Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to Mathematical and theoretical biology, mathematical biology and Integral equation, integral equations, being one of the ...
, is a
real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real ...
''V'' defined on the real line R with the following curious combination of properties: * ''V'' is
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
everywhere * The derivative ''V'' ′ is bounded everywhere * The derivative is not Riemann-integrable.


Definition and construction

The function is defined by making use of the
Smith–Volterra–Cantor set In mathematics, the Smith–Volterra–Cantor set (SVC), ε-Cantor set, or fat Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–V ...
and an infinite number or "copies" of sections of the function defined by :f(x) = \begin x^2 \sin(1/x), & x \ne 0 \\ 0, & x = 0.\end The construction of ''V'' begins by determining the largest value of ''x'' in the interval , 1/8for which ''f'' ′(''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 , 1/4 We then translate this function to the interval /8, 5/8so 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'' ′ is then considered on the smaller interval ,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

Volterra's function is differentiable everywhere just as ''f'' (as defined above) is. One can show that ''f'' ′(''x'') = 2''x'' sin(1/''x'') - cos(1/''x'') for ''x'' ≠ 0, which means that in any neighborhood of zero, there are points where ''f'' ′ takes values 1 and −1. Thus there are points where ''V'' ′ takes values 1 and −1 in every neighborhood of each of the endpoints of intervals removed in the construction of the
Smith–Volterra–Cantor set In mathematics, the Smith–Volterra–Cantor set (SVC), ε-Cantor set, or fat Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–V ...
''S''. In fact, ''V'' ′ is discontinuous at every point of ''S'', even though ''V'' itself is differentiable at every point of ''S'', with derivative 0. However, ''V'' ′ is continuous on each interval removed in the construction of ''S'', so the set of discontinuities of ''V'' ′ is equal to ''S''. Since the Smith–Volterra–Cantor set ''S'' has positive
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
, this means that ''V'' ′ is discontinuous on a set of positive measure. By Lebesgue's criterion for Riemann integrability, ''V'' ′ is not Riemann integrable. If one were to repeat the construction of Volterra's function with the ordinary measure-0 Cantor set ''C'' in place of the "fat" (positive-measure) Cantor set ''S'', one would obtain a function with many similar properties, but the derivative would then be discontinuous on the measure-0 set ''C'' instead of the positive-measure set ''S'', and so the resulting function would have a Riemann integrable derivative.


See also

*
Fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating its slopes, or rate of change at every point on its domain) with the concept of integral, inte ...


References


External links


''Wrestling with the Fundamental Theorem of Calculus: Volterra's function''
, talk by David Marius Bressoud
''Volterra's example of a derivative that is not integrable''
{{Webarchive, url=https://web.archive.org/web/20160303185034/http://www.macalester.edu/~bressoud/talks/apnc2004/Volterra.ppt , date=2016-03-03 (PPT), talk by David Marius Bressoud Fractals Measure theory General topology