132 3 Real Analysis
g(x)assumes both positive and negative values on this interval. Being continuous,g
has the intermediate value property, so there is somex 0 ∈( 0 , 1 )for whichg(x 0 )=0.
We have thus foundx 0 ∈( 0 , 1 )such thatf(x 0 )= 1 +^1 x 2
0
. The double inequality from the
statement follows from 2x 0 < 1 +x^20 < 1 +x 0 , which clearly holds since on the one
hand,x 02 − 2 x 0 + 1 =(x 0 − 1 )^2 >0, and on the other,x 02 <x 0.
Example.Prove that every continuous mapping of a circle into a line carries some pair
of diametrically opposite points to the same point.
Solution.Yes, this problem uses the intermediate value property, or rather the more
general property that the image through a continuous map of a connected set is connected.
The circle is connected, so its image must be an interval. This follows from a more
elementary argument, if we think of the circle as the gluing of two intervals along their
endpoints. The image of each interval is another interval, and the two images overlap,
determining an interval.
Identify the circle with the setS^1 ={z∈C||z|= 1 }.Iff :S^1 →Ris the
continuous mapping from the statement, thenψ:S^1 →R,ψ(z)=f(z)−f(− ̄z)is also
continuous (here the bar denotes the complex conjugate, and as such,− ̄zis diametrically
opposite toz).
Pickz 0 ∈S^1 .Ifψ(z 0 )=0, thenz 0 and−z 0 map to the same point on the line.
Otherwise,
ψ(−z 0 )=f(−z 0 )−f(z)=−ψ(z 0 ).Henceψtakes a positive and a negative value, and by the intermediate value property it
must have a zero. The property is proved.
A more difficult theorem of Borsuk and Ulam states that any continuous map of the
sphere into the plane sends two antipodal points on the sphere to the same point in the
plane. A nice interpretation of this fact is that at any time there are two antipodal points
on earth with the same temperature and barometric pressure.
We conclude our list of examples with a surprising fact discovered by Lebesgue.
Theorem.There exists a functionf :[ 0 , 1 ]→[ 0 , 1 ]that has the intermediate value
property and is discontinuous at every point.
Proof.Lebesgue’s function acts like an automaton. The value at a certain point is pro-
duced from information read from the digital expansion of the variable.
The automaton starts acting once it detects that all even-order digits have be-
come 0. More precisely, if x = 0 .a 0 a 1 a 2 ...,the automaton starts acting once
a 2 k =0 for allk ≥ n. It then reads the odd-order digits and produces the value
f(x)= 0 .a 2 n+ 1 a 2 n+ 3 a 2 n+ 5 ....If the even-order digits do not eventually become zero,