Nowhere analytic, infinitely differentiable function
Graph of the Fabius function on the interval [0,1].Extension of the function to the nonnegative real numbers.
In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius (1966). It was also written down as the Fourier transform of
by Børge Jessen and Aurel Wintner (1935).
The Fabius function is defined on the unit interval, and is given by the cumulative distribution function of
where the ξn are independent uniformly distributed random variables on the unit interval.
This function satisfies the initial condition , the symmetry condition for and the functional differential equation for It follows that is monotone increasing for with and
There is a unique extension of f to the real numbers that satisfies the same differential equation for all x. This extension can be defined by f (x) = 0 for x ≤ 0, f (x + 1) = 1 − f (x) for 0 ≤ x ≤ 1, and f (x + 2r) = −f (x) for 0 ≤ x ≤ 2r with r a positive integer. The sequence of intervals within which this function is positive or negative follows the same pattern as the Thue–Morse sequence.
mathematics, the Fabiusfunction is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius (1966). It was also...
As a counterexample see the Weierstrass function or the Fabiusfunction. Formally, a function f {\displaystyle f} is real analytic on an open set D {\displaystyle...
the Fabiusfunction. Although it might seem that such functions are the exception rather than the rule, it turns out that the analytic functions are scattered...
\quad |t|\geq 1\end{cases}}} with F ( t ) {\displaystyle F(t)} the Fabiusfunction. Dynamics of diabetes Epidemiology Population dynamics Classical electrodynamics...
1986. Fabius was 37 years old when he was appointed and is, after Gabriel Attal, the second youngest prime minister of the Fifth Republic. Fabius was also...
Marcus Fabius Quintilianus (Latin: [kᶣiːn.tɪ.li.ˈaː.nʊs]; c. 35 – c. 100 AD) was a Roman educator and rhetorician born in Hispania, widely referred to...
confirms the hypothesis of seafloor spreading at mid-ocean ridges. The Fabiusfunction is published. Chen Jingrun publishes Chen's theorem: every sufficiently...
Jewish Women's Studies & Gender Issues. Oktober 2008 16(16):174–195 Fabius Schach, Todesfallanzeige Ghetto Theresienstadt Fabius Schach, bei holocaust.cz...
Fabius Ambustus was a name used by ancient Roman men from a branch of the gens Fabia, including: Quintus Fabius Ambustus, consul 412 BC; son of Quintus...
Quintus Fabius Maximus Verrucosus, demanded Carthage choose between war and peace, to which his audience replied that Rome could choose. Fabius chose war...
Ceres (University of Texas Press, 1996), p. 36. Servius cites the historian Fabius Pictor (late 3rd century BC) as his source. Michael Lipka, Roman Gods: A...
victories credited to Gaius Poetelius Libo Visolus to "Κόιντος Φάβιος" (Quintus Fabius) instead. The fasti consulares list Gaius Sulpicius Longus as the dictator...
Quintus Fabius Maximus Verrucosus instead of Titus Manlius Torquatus, the senior ex-censor. Tuditanus justified his choice by saying that Fabius was the...
as Fabius did most patiently when warring against Hannibal, though many censured his delays; but when the time comes you must strike hard, as Fabius did...
Second Punic War with Carthage, Rome's earliest known annalists Quintus Fabius Pictor and Lucius Cincius Alimentus recorded history in Greek, and relied...
– Death Eater. Albert Runcorn – Ministry of Magic employee whose chief function was as an investigator of alleged Muggle-borns. Newt Scamander – Magizoologist...
finding himself at odds with the Catholic hierarchy, he left his directorial function at the JEC after being forced out in 1964 by Bishop Émile-Charles-Raymond...
might have to accept the yoke. A meeting of the senate was convened by Q. Fabius, the prefect of the city. He made such a violent attack upon the proposed...
panicked by these heavy defeats, appointed Quintus Fabius as dictator, with sole charge of the war effort. Fabius introduced the Fabian strategy of avoiding open...
strong national government requiring a national bank and foreign loans to function, while Jefferson believed the states and the farm element should primarily...
January 2013, Hermelin was appointed by Minister of Foreign Affairs Laurent Fabius as France's Special Representative for Economic Relations with India. In...
friend Nicomachus Flavianus Quintus Fabius Memmius Symmachus, aristocrat Symmachi–Nicomachi diptych Quintus Fabius Memmius Symmachus, his son, who edited...
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