Weierstrass Function


The pathological function


(originally defined for a=2) that is continuous but differentiable only on a set of points of measure zero. The plots above show f_a(x) for a=2 (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 f(x) 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 f contains points at which f 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 f has a finite derivative (namely, 1/2) at the set of points x=(2A+1)/(2B+1) where A and B are integers. Gerver (1971) then proved that f is not differentiable at any point of the form 2A/(2B+1) or (2A+1)/(2B). Together with the result of Hardy that f is not differentiable at any irrational value, this completely solved the problem of the differentiability f.

Amazingly, the value of f(x) can be computed exactly for rational numbers x=p/q as


See also

Blancmange Function, Continuous Function, Differentiable, Monsters of Real Analysis, Nowhere Differentiable Function

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Referenced on Wolfram|Alpha

Weierstrass Function

Cite this as:

Weisstein, Eric W. "Weierstrass Function." From MathWorld--A Wolfram Web Resource.

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