THE WEIERSTRASS PATHOLOGICAL FUNCTION
Until Weierstrass published his shocking paper in 1872, most of the mathematical world
(including luminaries like Gauss) believed that a continuous function could only fail to
be differentiable at some collection of isolated points. In fact, it turns out that “most”
continuous functions are nondifferentiable at
all
points. (To understand what this state
ment could mean, you should take courses in topology and measure theory.) However,
Weierstrass was not, in fact, the ﬁrst to construct such a pathological function. He was
preceded by Bolzano (in 1830), Cell’erier (also 1830), and Riemann (1862). None of the
others published their work (indeed, their examples were not discovered in their notes
until after their deaths).
All known examples of nondifferentiable continuous functions are constructed in a
similar fashion to the following example – they are limits of functions that oscillate more
and more on small scales, but with higherfrequency oscillations being damped quickly.
The example we give here is a faithful reproduction of Weierstrass’s original 1872 proof.
It is somewhat more complicated than the example given as Theorem 7.18 in Rudin, but
is superior in at least one important way, as explained in Remark 2.
Theorem 1.
Let
0
< a <
1
, and choose a positive odd integer
b
large enough that
π
ab

1
<
2
3
(i.e.
ab >
1 +
3
π
2
). For example, take
a
=
1
2
and
b
= 11
. Deﬁne the function
W
:
R
→
R
by
W
(
x
) =
∞
X
n
=0
a
n
cos(
b
n
πx
)
.
Then
W
is uniformly continuous on
R
, but is differentiable at no point.
Remark
2
.
Notice that the partial sums
W
N
(
x
) =
∑
N
n
=0
a
n
cos(
b
n
πx
)
are all
C
∞
functions.
As the following proof shows, these partial sums converge uniformly to
W
, and so we
have an example here of a sequence of
C
∞
functions that converge uniformly to a nowhere
differentiable function. This is the most dramatic demonstration that differentiability is
not
preserved under uniform convergence! The example of Theorem 7.18 in Rudin, while
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 Fall '10
 Prof.KatrinWehrheim
 Calculus, Logic, Derivative, Continuous function, Uniform convergence

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