9.4 *Two Results of Weierstrass 569
AN EVERYWHERE CONTINUOUS,
NOWHERE DIFFERENTIABLE FUNCTION
In 1872 , Weierstrass presented a paper to the Academy in Berlin in which
he showed that the function^9
00
f(x) = L bk cos(ak7rx), (12)
k=l
where a is an odd natural number, b E [O, 1), and ab > 1 +^3 {, is continuous
everywhere on ( -oo, oo) yet is differentiable nowhere. This example created a
sensation in the mathematical world because it challenged the then-widespread
belief among mathematicians that continuous functions had to be differentiable
everywhere except possibly at isolated "singular" points. In fact, A. Ampere
had even published a "proof' of this "fact" in 1806. Weierstrass' example thus
brought to an end a long string of futile attempts to show that differentiabil-
ity somehow follows from continuity. It is perhaps even more remarkable that
each term of the series (12) is infinitely differentiable everywhere, yet the sum
function is not differentiable anywhere.
Weierstrass was not the first to claim the existence of an everywhere con-
tinuous, nowhere differentiable function, but he was the first to provide a rig-
orous proof. About 1830 , Bolzano had constructed a similar example but was
unable to prove that it was nowhere differentiable. Bolzano was primarily a
philosopher-theologian rather than a mathematician, and lived in Prague, which
was not a major center of mathematical research. Although he did not publish
his example, it has subsequently been proved correct.
In 1860 , the Swiss mathematician C. Cellerier gave another example,
f(x ) = f sin(~kx)'
n=l a
(13)
which is nowhere differentiable when a is a sufficiently large positive integer.
His result was not published, however, until 1890, whereas Weierstrass' result
was published in 1875.
According to Weierstrass, Riemann claimed in his lectures in 1861 that the
function
f(x) = f sin~~2x)'
k=l
(14)
is nowhere differentiable, or at least nondifferentiable on a dense subset of
R Weierstrass was una ble to verify nondifferentia bility of Riemann's function
- The historica l notes concerning this function a re d erived from Chapter 6 of [66], Section
6.4 of [21], Section IIl.9 of [61], pages 955-6 of [75], and pages 44- 47 of [62], which are all
recommended for further reading.