1549901369-Elements_of_Real_Analysis__Denlinger_

570 Chapter 9 • Sequences and Series of Functions

(14) and subsequently constructed his own example, the function (12) above.
In 1970 , J. Gerver proved that, in fact, Riemann's function (14) is differentiable
at infinitely many points; namely, when x = a7r, where a = ~"!:${,form, n EN.
The function we a re going to consider is a clever simplification of Weier-
strass' example, published by Davidson and Donsig in [31]:^10

f(x) = f cos(~~k7rx). (15)

k=l

The graph of the sum of the first three terms is shown in Figure 9.12 below.
The idea b ehind (15) is to use a uniformly convergent series of continu-
ous functions to ensure that the limit function is continuous, and use cosine
functions of smaller and smaller periods as summands to ensure that the limit
function has infinitely many oscillations in every open interval, causing it to be
nowhere differentia ble.

-0.75

"-.......

y=

cos(!Onx)

2

y

I

cos(IOOnx)

+ +

4

Figure 9.12

cos(IOOOnx)

8

x

Theorem 9.4.1 (Weierstrass): There exists a Junction that is continuous
everywhere on ( -oo, oo), but differentiable nowhere.

10. Proofs using Weierstrass' function may b e found in [63] as Theorem (17.7) , in Chapter
    9 of [34], and in Section 6.4 of [21]. Most authors use a n entirely different kind of function,
    based on one devised by B. L. van der Waerden in 19 30. See Example 8.13 of [30] for details.