The pathological function

(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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