(originally defined for ) that is continuous
but differentiable only on a set of points of
measure zero. The plots above show for (red), 3 (green), and 4 (blue).
The function was published by Weierstrass but, according to lectures and writings by Kronecker and Weierstrass, Riemann seems to have claimed already in 1861 that
the function
is not differentiable on a set dense in the reals. However, Ullrich (1997) indicates
that there is insufficient evidence to decide whether Riemann actually bothered to
give a detailed proof for this claim. du Bois-Reymond (1875) stated without proof
that every interval of contains points at which does not have a finite derivative, and Hardy (1916) proved
that it does not have a finite derivative at any irrational and some of the rational
points. Gerver (1970) and Smith (1972) subsequently proved that has a finite derivative (namely, 1/2) at the set of points
where
and
are integers. Gerver (1971) then proved that is not differentiable at any point of the form or . Together with the result of Hardy that is not differentiable at any irrational value, this completely
solved the problem of the differentiability .
Amazingly, the value of can be computed exactly for rational numbers as
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) that is continuous
but differentiable only on a set of points of
measure zero. The plots above show 
for 
(red), 3 (green), and 4 (blue).
is not differentiable on a set dense in the reals. However, Ullrich (1997) indicates
that there is insufficient evidence to decide whether Riemann actually bothered to
give a detailed proof for this claim. du Bois-Reymond (1875) stated without proof
that every interval of 
contains points at which 
does not have a finite derivative, and Hardy (1916) proved
that it does not have a finite derivative at any irrational and some of the rational
points. Gerver (1970) and Smith (1972) subsequently proved that 
has a finite derivative (namely, 1/2) at the set of points
where 
and 
are integers. Gerver (1971) then proved that 
is not differentiable at any point of the form 
or 
. Together with the result of Hardy that 
is not differentiable at any irrational value, this completely
solved the problem of the differentiability 
.
can be computed exactly for rational numbers 
as
."
Amer. J. Math. 92, 33-55, 1970.Gerver, J. "More on
the Differentiability of the Riemann Function." Amer. J. Math. 93,
33-41, 1971.Girgensohn, R. "Functional Equations and Nowhere Differentiable
Functions." Aeq. Math. 46, 243-256, 1993.Gluzman,
S. and Sornette, D. "Log-Periodic Route to Fractal Functions." 4 Oct 2001.
einer stetigen, nicht differenzierbaren
Funktion." Results Math. 31, 245-265, 1997.Volkert,
K. "Die Geschichte der pathologischen Funktionen--Ein Beitrag zur Entstehung
der mathematischen Methodologie." Arch. Hist. Exact Sci. 37, 193-232,
1987.Weierstrass, K. Abhandlungen aus der Functionenlehre. Berlin:
J. Springer, p. 97, 1886.