Advanced book on Mathematics Olympiad

3.2 Continuity, Derivatives, and Integrals 129

Summing up, we obtain

f(x)−f

(x

2 n

)

=x

(

1

2

+

1

4

+···+

1

2 n

)

,

which, whenntends to infinity, becomesf(x)− 1 =x. Hencef(x)=x+1isthe
(unique) solution. 

We will now present the spectacular example of a continuous curve that covers a
square completely. A planar curveφ(t) =(x(t), y(t))is called continuous if both
coordinate functionsx(t)andy(t)depend continuously on the parametert.

Peano’s theorem.There exists a continuous surjectionφ:[ 0 , 1 ]→[ 0 , 1 ]×[ 0 , 1 ].

Proof.G. Peano found an example of such a function in the early twentieth century. The
curve presented below was constructed later by H. Lebesgue.
The construction of this “Peano curve’’ uses the Cantor set. This is the setCof all
numbers in the interval[ 0 , 1 ]that can be written in base 3 with only the digits 0 and

2. For example, 0.1isinCbecause it can also be written as 0. 0222 ...,but 0.101 is
   not. The Cantor set is obtained by removing from[ 0 , 1 ]the interval(^13 ,^23 ), then(^19 ,^29 )
   and(^79 ,^89 ), then successively from each newly formed closed interval an open interval
   centered at its midpoint and^13 of its size (Figure 19). The Cantor set is afractal: each
   time we cut a piece of it and magnify it, the piece resembles the original set.

Figure 19

Next, we define a functionφ:C→[ 0 , 1 ]×[ 0 , 1 ]in the following manner. For a
number written in base 3 as 0.a 1 a 2 ...an...with only the digits 0 and 2 (hence in the Can-
tor set), divide the digits by 2, then separate the ones in even positions from those in odd
positions. Explicitly, ifbn=a 2 n,n≥1, construct the pair( 0 .b 1 b 3 b 5 ..., 0 .b 2 b 4 b 6 ...).
This should be interpreted as a point in[ 0 , 1 ]×[ 0 , 1 ]with coordinates written in base

2. Thenφ( 0 .a 1 a 2 a 3 a 4 ...)=( 0 .b 1 b 3 ..., 0 .b 2 b 4 ...). The function is clearly onto. Is it
   continuous?
   First, what does continuity mean in this case? It means that whenever a sequence
   (xn)ninCconverges to a pointx∈C, the sequence(φ(xn))nshould converge toφ(x).
   Note that since the complement ofCis a union of open intervals,Ccontains all its limit
   points. Moreover, the Cantor set has the very important property that a sequence(xn)n
   of points in it converges tox ∈Cif and only if the base-3 digits ofxnsuccessively
   become equal to the digits ofx. It is essential that the base-3 digits of a number inC
   can equal only 0 or 2, so that the ambiguity of the ternary expansion is eliminated. This
   fundamental property of the Cantor set guarantees the continuity ofφ.