As mentioned in the preceding chapter, if a function f : R → R
is nowhere differentiable, then f is nowhere monotone, i.e., there does not exist a nondegenerate subinterval of R on which f is monotone.
This chapter is devoted to some constructions of functions also acting from R into R, differentiable everywhere but nowhere monotone. The ques- tion of the existence of such functions is obviously typical for classical math- ematical analysis. And it should be noticed that many mathematicians of the end of the 19th century and of the beginning of the 20th century tried to present various constructions of the above-mentioned functions. As a rule, their constructions were either incorrect or, at least, incomplete. As pointed out in [83], the first explicit construction of such a function was given by K¨opcke in 1889. Another example was suggested by Pereno in 1897 (this example is presented in [74], pp. 412–421). In addition, Denjoy gave in his extensive work [48] a proof of the existence of an everywhere differentiable nowhere monotone function, as a consequence of his deep investigations concerning trigonometric series and their convergence.
Afterwards, a number of distinct proofs of the existence of such functions were given by several authors (see, e.g., [65], [83], [126], [229]).
We begin with the discussion of the construction presented in [83]. This construction is completely elementary and belongs to classical mathematical analysis. We need some easy auxiliary propositions.
Lemma 1. Let r and s be two strictly positive real numbers. The following assertions hold:
(1) if r > s, then
(r − s)/(r2− s2) < 2/r; (2) if r > 1 and s > 1, then
Proof. Indeed, we have
(r − s)/(r2− s2) = 1/(r + s) < 1/r < 2/r.
Thus, (1) is valid. Further, it can easily be checked that the inequality of (2) is equivalent to the inequality
(r − s)2+ (r − 1)(s − 1) + r2+ r + 3s > 5
which, obviously, is true under our assumptions r > 1 and s > 1. This completes the proof ofLemma 1.
Lemma 2. Let φ be the function from R into R defined by φ(x) = (1 + |x|)−1/2 (x ∈ R)
and let a and b be any two distinct real numbers. Then we have
(1/(b − a)) Z b
a
φ(x)dx < 4min(φ(a), φ(b)).
Proof. Without loss of generality we may assume that a < b. Only three cases are possible.
1. 0 ≤ a < b. In this case, taking into account (1) of Lemma 1, we can write (1/(b − a)) Z b a φ(x)dx = 2((1 + b)1/2− (1 + a)1/2)/((1 + b) − (1 + a)) < 4/(1 + b)1/2= 4min(φ(a), φ(b)).
2. a < b ≤ 0. This case can be reduced to the previous one, because of the evenness of our function φ.
3. a < 0 < b. In this case, taking into account (2) of Lemma 1, we can write (1/(b − a)) Z b a φ(x)dx = 2((1 + b)1/2+ (1 − a)1/2− 2)/((1 + b) + (1 − a) − 2) < 4min(φ(a), φ(b)). This ends the proof of Lemma 2.
everywhere differentiable nowhere monotone functions 123
Lemma 3. Let n > 0 be a natural number and let ψ : R → R
be any function of the form ψ(x) = X
1≤k≤n
ckφ(dk(x − tk)) (x ∈ R),
where φ is the function fromLemma 2,
c1, c2, . . . , cn,
d1, d2, . . . , dn
are strictly positive real numbers, and t1, t2, . . . , tn
are real numbers. Then
(1/(b − a)) Z b
a
ψ(x)dx < 4min(ψ(a), ψ(b))
for all distinct reals a and b.
Proof. The assertion follows immediately from Lemma 2, by taking into account the fact that
(1/(b − a)) Z b a φ(d(x − t))dx = (1/(d(b − t) − d(a − t))) Z d(b−t) d(a−t) φ(x)dx
for any d > 0 and t ∈ R.
Lemma 4. Let (ψn)n≥1 be a sequence of functions as in Lemma 3. For
any point x ∈ R and for each integer n ≥ 1, let us define
Ψn(x) =
Z x
0
ψn(z)dz
and suppose that, for some a ∈ R, the series P
n≥1ψn(a) is convergent. Denote X n≥1 ψn(a) = s < +∞. Then we have:
(1) the series F (x) =P
n≥1Ψn(x) converges uniformly on every bounded
subinterval of R;
(2) the function F is differentiable at the point a and F0(a) = s.
In particular, if
X
n≥1
ψn(z) = f (z) < +∞
for each z ∈ R, then the function F is differentiable everywhere on R and the equality
F0 = f holds true.
Proof. Take any b ∈ R satisfying the relation b ≥ |a|. In view of Lemma 3, for all points x ∈ [−b, b] and for all integers n ≥ 1, we may write
|Ψn(x)| ≤ | Z a 0 ψn(z)dz| + | Z x a ψn(z)dz| ≤
4|a|ψn(a) + 4|x − a|ψn(a) ≤ 12bψn(a).
This shows the uniform convergence of X
n≥1
Ψn(x) (x ∈ [−b, b])
on the segment [−b, b].
Further, let ε > 0 be given. Pick a natural number k > 0 such that
10 ·X
n>k
ψn(a) < ε.
Because all functions ψn are continuous on the whole R (in particular, at
a), there exists some δ > 0 such that
|(1/h) Z a+h a ψn(z)dz − ψn(a)| < ε/2k, whenever 0 < |h| < δ, 1 ≤ n ≤ k.
everywhere differentiable nowhere monotone functions 125
Consequently, assuming that 0 < |h| < δ and applyingLemma 3again, we get |(F (a + h) − F (a))/h − s| = |X n≥1 ((1/h) Z a+h a ψn(z)dz − ψn(a))| ≤ X 1≤n≤k |(1/h) Z a+h a ψn(z)dz − ψn(a)| + X n>k ((1/h) Z a+h a ψn(z)dz + ψn(a)) < ε/2 +X n>k 5ψn(a) < ε.
We thus conclude that
F0(a) = s, and the lemma is proved.
Lemma 5. Let n > 0 be a natural number, let I1, ..., In be pairwise
disjoint nondegenerate segments on R and let tk denote the midpoint of Ik
for each natural number k ∈ [1, n]. Fix any strictly positive real numbers ε, y1, . . . , yn.
Then there exists a function ψ as in Lemma 3 such that, for all natural numbers k ∈ [1, n], the following relations are fulfilled:
(1) ψ(tk) > yk;
(2) (∀x ∈ Ik)(ψ(x) < yk+ ε);
(3) (∀x ∈ R \ (I1∪ ... ∪ In))(ψ(x) < ε).
Proof. Let us denote
ck= yk+ ε/2 (1 ≤ k ≤ n).
Then, for each integer k ∈ [1, n], define
φk(x) = ckφ(dk(x − tk)) (x ∈ R),
where dk> 0 is chosen so large that
(∀x ∈ R \ Ik)(φk(x) < ε/2n).
Finally, put
Then, taking into account the fact that
maxx∈Rφk(x) = φk(tk) = ck (1 ≤ k ≤ n),
it is easy to check that the function ψ satisfies relations (1), (2) and (3). Lemma 6. Let any two disjoint countable subsets
{tk : k ∈ N, k ≥ 1},
{rk : k ∈ N, k ≥ 1}
of R be given. Then there exists a function F : R → R such that: (1) F is differentiable everywhere on R; (2) 0 < F0(x) ≤ 1 for all x ∈ R; (3) F0(tk) = 1 for each k ∈ N \ {0}; (4) F0(rk) < 1 for each k ∈ N \ {0}.
Proof. We are going to construct by recursion the sequence (ψn)n≥1of
functions as inLemma 3with some additional properties. Namely, denoting fn =
X
1≤k≤n
ψk,
we wish the following conditions to be satisfied:
(i) for any integer n ≥ 1 and for all integers k ∈ [1, n], we have fn(tk) > 1 − 1/n;
(ii) for any integer n ≥ 1 and for each point x ∈ R, we have fn(x) < 1 − 1/(n + 1);
(iii) for any integer n ≥ 1 and for all integers k ∈ [1, n], we have ψn(rk) < 1/(2n · 2n).
First, choose a nondegenerate segment I1with midpoint t1, such that r16∈ I1
and applyLemma 5with
everywhere differentiable nowhere monotone functions 127
Evidently, we obtain ψ1 and f1 = ψ1, such that relations (i), (ii) and (iii)
are fulfilled for n = 1.
Suppose now that, for a natural number n > 1, we have already defined the functions
ψ1, . . . , ψk, . . . , ψn−1
satisfying the corresponding analogues of (i) - (iii) for n − 1. Pick disjoint nondegenerate segments I1, ... , In in such a way that:
(a) tk is the midpoint of Ik for each integer k ∈ [1, n];
(b) Ik∩ {r1, ... , rn} = ∅ for each integer k ∈ [1, n];
(c) for any integer k ∈ [1, n] and for any point x ∈ Ik, we have the
inequality
fn−1(x) < fn−1(tk) + δ,
where
δ = 1/(n(n + 1)) − 1/(2n · 2n). We now can applyLemma 5with
ε = 1/(2n · 2n) and
yk = 1 − (1/n) − fn−1(tk) (1 ≤ k ≤ n).
Applying the above-mentioned lemma, we get the function ψn. Clearly,
ψn(rk) < ε = 1/(2n · 2n)
for all natural numbers k ∈ [1, n], so relation (iii) holds true. Further, for any natural number k ∈ [1, n], we also have
fn(tk) = fn−1(tk) + ψn(tk) > fn−1(tk) + yk= 1 − 1/n,
which shows that relation (i) holds true, too. Finally, in order to verify (ii), fix any point x ∈ R. If, for some integer k ∈ [1, n], the point x belongs to Ik, then we may write
fn(x) = fn−1(x) + ψn(x) < fn−1(tk) + δ + yk+ ε =
1 − 1/n + 1/(n(n + 1)) = 1 − 1/(n + 1). If x does not belong to I1∪ ... ∪ In, then
Thus, relation (ii) holds true, too. Proceeding in this manner, we are able to construct the required sequence (ψn)n≥1. Putting
f =X n≥1 ψn= limn→+∞fn and F (x) = Z x 0 f (z)dz (x ∈ R), we obtain the function
F : R → R. In view ofLemma 4, we also get
F0(x) = f (x) (x ∈ R). Further, the definition of F immediately implies
0 < F0(x) ≤ 1 (x ∈ R), F0(tk) = 1 (k ∈ N, k ≥ 1).
Now, fix an integer k ≥ 1 and let n be a natural number strictly greater than k. Then F0(rk) = fn−1(rk) + X m≥n ψm(rk) < 1 − 1/n + X m≥n 1/(2m · 2m) < 1 − 1/n + 1/2n = 1 − 1/2n < 1.
This completes the proof ofLemma 6. Theorem 1. There exists a function
H : R → R such that:
(1) H is differentiable everywhere on R; (2) H0 is bounded on R;
(3) H is monotone on no nondegenerate subinterval of R. Proof. Denote by
{tn : n ∈ N, n > 0},
everywhere differentiable nowhere monotone functions 129
any two disjoint countable everywhere dense subsets of R. Using the result of the previous lemma, take any two everywhere differentiable functions
F : R → R, G : R → R, satisfying the relations:
(a) 0 < F0(x) ≤ 1 and 0 < G0(x) ≤ 1 for all x ∈ R;
(b) F0(tn) = 1 and F0(rn) < 1 for each natural number n ≥ 1;
(c) G0(rn) = 1 and G0(tn) < 1 for each natural number n ≥ 1.
Now, define
H = F − G. Obviously, we have
H0(tn) > 0, H0(rn) < 0 (n ∈ N, n ≥ 1).
Because both the sets
{tn : n ∈ N, n > 0},
{rn : n ∈ N, n > 0}
are everywhere dense in R, we infer that H cannot be monotone on any subinterval of R. Also, the relation
−1 < H0(x) < 1 (x ∈ R)
implies that H0 is bounded, and the theorem has thus been proved. In fact, the preceding argument establishes the existence of many func- tions
f : R → R,
which are everywhere differentiable, nowhere monotone and such that f0 is bounded on R. Let us mention some other interesting and unusual proper- ties of any such function f .
1. f has a point of a local maximum and a point of a local minimum in every nonempty open subinterval of R. Actually, for each nondegenerate segment [a, b] ⊂ R, we can find some points x1and x2satisfying the relations
a < x1< x2< b,
Let us denote
M = supt∈[x1,x2] f (t).
Then, for some τ ∈ [x1, x2], we must have
f (τ ) = M,
and it is clear that τ must be in the interior of [x1, x2]. In other words, τ is
a point of a local maximum for our f .
Applying a similar argument, we can also find a point of a local minimum of f on the same nondegenerate segment [a, b].
2. Because f0is bounded, the function f satisfies the so-called Lipschitz condition, i.e., for some constant d ≥ 0, we have
|f (x) − f (y)| ≤ d|x − y| (x ∈ R, y ∈ R). Note that, in the latter relation, we may put
d = supt∈R|f0(t)|.
In particular, f is absolutely continuous. This also implies that f0 is
Lebesgue integrable on each bounded subinterval of R.
3. The function f0 is not integrable in the Riemann sense on any non- degenerate segment [a, b] ⊂ R. To see this, suppose otherwise, i.e., suppose that f0 is Riemann integrable on [a, b]. Then, according to a well-known theorem of mathematical analysis, f0must be continuous at almost all (with respect to the standard Lebesgue measure) points of [a, b] (see, e.g., [154]). Taking into account the fact that f0 changes its sign on each nonempty open subinterval of R, we infer that f0 must be zero at almost all points of [a, b]. Consequently, f must be constant on [a, b], which is impossible. The contradiction obtained yields the desired result.
4. Being a derivative, the function f0 belongs to the first Baire class, i.e., it can be represented as a pointwise limit of a sequence of continuous functions (seeExample 1 from Chapter 2). Hence, by virtue of the classical Baire theorem (see, e.g., [120], [154] orTheorem 3 from Chapter 2), the set of all those points of R at which f0 is continuous is residual (co-meager), i.e., is the complement of a first category subset of R.
5. Let us denote
X = {t ∈ R : f0(t) > 0}, Y = {t ∈ R : f0(t) < 0}.
everywhere differentiable nowhere monotone functions 131
The sets X and Y are disjoint, Lebesgue measurable and have the property that, for each nondegenerate segment [a, b] ⊂ R, the relations
λ(X ∩ [a, b]) > 0, λ(Y ∩ [a, b]) > 0
are fulfilled (where λ denotes, as usual, the Lebesgue measure on R). In order to demonstrate this fact, suppose, for example, that
λ(Y ∩ [a, b]) = 0.
Then we get f0(t) ≥ 0 for almost all points t ∈ [a, b]. But this immediately implies that our function f , being of the form
f (x) = Z x
0
f0(t)dt + f (0) (x ∈ R),
is increasing on [a, b], which contradicts the definition of f .
Exercise 1. Give a direct construction of two disjoint Lebesgue mea- surable subsets X and Y of R, such that, for any nonempty open interval I ⊂ R, the inequalities
λ(I ∩ X) > 0, λ(I ∩ Y ) > 0
hold true. More generally, show that there exists a countable partition {Xn : n < ω}
of R consisting of Lebesgue measurable sets and such that, for any nonempty open subinterval I of R and for any natural number n, the relation
λ(I ∩ Xn) > 0
is fulfilled.
Exercise 2. Denote by E the family of all Lebesgue measurable subsets of the unit segment [0, 1]. For any two sets X ∈ E and Y ∈ E , put
d(X, Y ) = λ(X4Y ).
Identifying all those X and Y , for which d(X, Y ) = 0, we come to the metric space (E , d). Check that (E , d) is complete and separable, i.e., is a Polish
space. Further, let E0 be a subspace of E consisting of all sets X ∈ E such that
λ(X ∩ I) > 0, λ(([0, 1] \ X) ∩ I) > 0
for each nondegenerate subinterval I of [0, 1]. Show that E0 is the comple- ment of a first category subset of E . Hence, according to the Baire theorem, we have
E06= ∅.
It is useful to compare this fact withExercise 1 above.
Now, we are going to present an essential generalization ofTheorem 1
due to Weil (see [229]). Namely, Weil gave a proof of the existence of every- where differentiable nowhere monotone functions (with a bounded deriva- tive) by using the above-mentioned Baire theorem.
We recall that a function
f : R → R
is a derivative if there exists at least one everywhere differentiable function F : R → R
satisfying the relation
(∀x ∈ R)(F0(x) = f (x)). Let us consider the set
D = {f : f is a derivative and f is bounded}.
Obviously, D is a vector space over R. We may equip this set with a metric d defined by the formula
d(f, g) = supx∈R|f (x) − g(x)|.
Clearly, the metric d produces the topology of uniform convergence. In view of the well-known theorem of analysis, a uniform limit of a sequence of bounded derivatives is a bounded derivative (cf. Exercise 20 from Chapter 2). This shows, in particular, that the pair (D, d) is a Banach space (it can easily be seen that it is nonseparable). Take any function f ∈ D and consider the set f−1(0). We assert that this set is a Gδ-subset of R. Indeed,
we may write
everywhere differentiable nowhere monotone functions 133 where a function F : R → R is such that F0(x) = f (x) (x ∈ R). Equivalently, we have f−1(0) = \ 1≤n<ω ( [ n≤m<ω {x ∈ R : m(F (x + 1/m) − F (x)) < 1/n}).
This formula yields at once the desired result. Let us put
D0= {f ∈ D : f−1(0) is everywhere dense in R}.
We need the following simple fact.
Lemma 7. The set D0 is a closed vector subspace of the space D.
Consequently, D0 is a Banach space, as well.
Proof. First, we show that D0 is closed in D. Let {fk : k < ω} be
a sequence of functions from D0, converging (in metric d) to some function
f ∈ D. We put
Zk = fk−1(0) (k < ω).
Then all the sets Zk are everywhere dense Gδ-subsets of R. Therefore, the
set
Z = ∩{Zk : k < ω}
is an everywhere dense Gδ-subset of R, too. Obviously, the inclusion
Z ⊂ f−1(0) is valid. Thus, we obtain that f ∈ D0.
Now, let us demonstrate that D0 is a vector subspace of D. Clearly, if
f ∈ D0 and t ∈ R, then tf ∈ D0. Further, take any two functions g ∈ D0
and h ∈ D0 and consider the sets
Zg = g−1(0),
Zh= h−1(0).
Then the set Zg∩ Zh is an everywhere dense Gδ-subset of R, and it is
evident that
where
Zg+h= (g + h)−1(0).
This shows that D0 is a vector space.
Lemma 7has thus been proved.
Notice now that the space D0 is nontrivial, i.e., contains nonzero func-
tions. For instance, this fact follows directly fromTheorem 1. But it can also easily be proved by another argument.
Exercise 3. Give a direct proof (i.e., without the aid of Theorem 1) that D0 contains nonzero elements.
Theorem 2. Let us denote
E = {f ∈ D0 : there is a nondegenerate subinterval of R
on which f preserves its sign}. Then the set E is of first category in the space D0.
Proof. Let {In : n ∈ N} be the family of all subintervals of R with
rational end-points. For each n ∈ N, put
An= {f ∈ D0 : (∀x ∈ In)(f (x) ≥ 0)},
Bn= {f ∈ D0 : (∀x ∈ In)(f (x) ≤ 0)}.
Clearly, we have
E = ∪n∈N(An∪ Bn),
so it suffices to demonstrate that each of the sets An and Bn is closed
and nowhere dense. We shall establish this fact only for An (for Bn, the
argument is analogous). The closedness of An is trivial. In order to prove
that Ancontains no ball in D0, take any f ∈ D0and fix an arbitrary ε > 0.
Because f ∈ D0, there exists a point x ∈ In such that
f (x) = 0.
Now, by starting with the existence of a nonzero bounded derivative be- longing to D0, it is easy to show that there is a function h ∈ D0 for which
h(x) < 0, supy∈R|h(y)| < ε.
Let us define
everywhere differentiable nowhere monotone functions 135
Then the function g belongs to the ball of D0 with center f and radius ε.
At the same time, g does not belong to An because
g(x) = f (x) + h(x) = h(x) < 0.
This establishes that An is nowhere dense in D0, and the theorem has thus
been proved.
In the next chapter, we shall consider one more proof of the existence of everywhere differentiable nowhere monotone functions, by applying some properties of the so-called density topology on R.
Exercise 4. Let E be an arbitrary topological space. Recall that a family N of subsets of E is a net in E if each open subset of E can be represented as the union of some subfamily of N (recall also that the concept of a net, for topological spaces, was introduced by Archangelskii; obviously, it generalizes the concept of a base of a topological space). We denote by the symbol nw(E) the smallest cardinality of a net in E.
Let now
f : E → R be a function. We put:
lmaxv(f ) = the set of all those t ∈ R for which there exists a nonempty open subset U of E such that t = sup(f (U )) and, in addition, there is a point e ∈ U such that f (e) = sup(f (U ));
lminv(f ) = the set of all those t ∈ R for which there exists a nonempty open subset V of E such that t = inf (f (V )) and, in addition, there is a point e ∈ V such that f (e) = inf (f (V )).
Check that:
card(lmaxv(f )) ≤ nw(E) + ω, card(lminv(f )) ≤ nw(E) + ω.
In particular, if E possesses a countable net, then the above-mentioned subsets of R are at most countable.
Denote also:
slmax(f ) = the set of all points e ∈ E having the property that there exists a neighborhood U (e) such that f (e) > f (x) for each x ∈ U (e);
slmin(f ) = the set of all points e ∈ E having the property that there exists a neighborhood V (e) such that f (e) < f (x) for each x ∈ V (e).
Verify that:
card(slmax(f )) ≤ nw(E) + ω, card(slmin(f )) ≤ nw(E) + ω.
In particular, if E possesses a countable net, then the sets slmax(f ) and slmin(f ) are at most countable.
Finally, for E = R, give an example of a continuous function f for which the latter two sets are everywhere dense in E.