In mathematical analysis, a Pompeiu derivative is a real-valued function of one real variable that is the derivative of an everywhere differentiable function and that vanishes in a dense set. In particular, a Pompeiu derivative is discontinuous at any point where it is not 0. Whether non-identically zero such functions may exist was a problem that arose in the context of early-1900s research on functional differentiability and integrability. The question was affirmatively answered by Dimitrie Pompeiu by constructing an explicit example; these functions are therefore named after him.
Pompeiu's construction is described here. Let
x
3
denote the real cubic root of the real number
x
.
Let
{
q
j
}
j
∈
N
be an enumeration of the rational numbers in the unit interval
[
0
,
1
]
.
Let
{
a
j
}
j
∈
N
be positive real numbers with
∑
j
a
j
<
∞
.
Define, for all
x
∈
[
0
,
1
]
g
(
x
)
:=
∑
j
=
0
∞
a
j
x
−
q
j
3
.
Since for any
x
∈
[
0
,
1
]
each term of the series is less than or equal to aj in absolute value, the series uniformly converges to a continuous, strictly increasing function g(x), due to the Weierstrass M-test. Moreover, it turns out that the function g is differentiable, with
g
′
(
x
)
:=
1
3
∑
j
=
0
∞
a
j
(
x
−
q
j
)
2
3
>
0
,
at any point where the sum is finite; also, at all other points, in particular, at any of the
q
j
,
one has
g
′
(
x
)
:=
+
∞
.
Since the image of
g
is a closed bounded interval with left endpoint
0
=
g
(
0
)
,
up to a multiplicative constant factor one can assume that g maps the interval
[
0
,
1
]
onto itself. Since g is strictly increasing, it is a homeomorphism; and by the theorem of differentiation of the inverse function, its composition inverse
f
:=
g
−
1
has a finite derivative at any point, which vanishes at least in the points
{
g
(
q
j
)
}
j
∈
N
.
These form a dense subset of
[
0
,
1
]
(actually, it vanishes in many other points; see below).
It is known that the zero-set of a derivative of any everywhere differentiable function is a Gδ subset of the real line. By definition, for any Pompeiu function this set is a dense Gδ set, therefore by the Baire category theorem it is a residual set. In particular, it possesses uncountably many points.
A linear combination af(x) + bg(x) of Pompeiu functions is a derivative, and vanishes on the set {f = 0} ∩ {g = 0}, which is a dense Gδ by the Baire category theorem. Thus, Pompeiu functions are a vector space of functions.
A limit function of a uniformly convergent sequence of Pompeiu derivatives is a Pompeiou derivative. Indeed, it is a derivative, due to the theorem of limit under the sign of derivative. Moreover, it vanishes in the intersection of the zero sets of the functions of the sequences: since these are dense Gδ sets, the zero set of the limit function is also dense.
As a consequence, the class E of all bounded Pompeiu derivatives on an interval [a, b] is a closed linear subspace of the Banach space of all bounded functions under the uniform distance (hence, it is a Banach space).
Pompeiu's above construction of a positive function is a rather peculiar example of a Pompeiu's function: a theorem of Weil states that generically a Pompeiu derivative assumes both positive and negative values in dense sets, in the precise meaning that such functions constitute a residual set of the Banach space E.
