Advanced book on Mathematics Olympiad

3.2 Continuity, Derivatives, and Integrals 131

390.Letaandbbe real numbers in the interval( 0 ,^12 )and letfbe a continuous real-

valued function such that

f(f(x))=af (x)+bx, for allx∈R.

Prove thatf( 0 )=0.

391.Letf:[ 0 , 1 ]→Rbe a continuous function. Prove that the series

∑∞

n= 1

f(xn)

2 n is

convergent for everyx∈[ 0 , 1 ]. Find a functionfsatisfying

f(x)=

∑∞

n= 1

f(xn)

2 n

, for allx∈[ 0 , 1 ].

392.Prove that there exists a continuous surjective functionψ:[ 0 , 1 ]→[ 0 , 1 ]×[ 0 , 1 ]

that takes each value infinitely many times.

393.Give an example of a continuous function on an interval that is nowhere differen-

tiable.

3.2.3 The Intermediate Value Property............................

A real-valued functionfdefined on an interval is said to have the intermediate value
property (or the Darboux property) if for everya<bin the interval and for everyλ
betweenf(a)andf(b), there existscbetweenaandbsuch thatf(c)=λ. Equivalently,
a real-valued function has the intermediate property if it maps intervals to intervals. The
higher-dimensional analogue requires the function to map connected sets to connected
sets. Continuous functions and derivatives of functions are known to have this property,
although the class of functions with the intermediate value property is considerably larger.
Here is a problem from the 1982 Romanian Mathematical Olympiad, proposed by
M. Chiri ̧ta. ̆

Example.∫ Letf :[ 0 , 1 ]→R be a continuous function with the property that

1

0 f(x)dx=

π

4. Prove that there existsx^0 ∈(^0 ,^1 )such that

1

1 +x 0

<f(x 0 )<

1

2 x 0

.

Solution.Note that

∫ 1

0

1

1 +x^2

dx=

π

4

.

Consequently, the integral of the functiong(x)=f(x)− 1 +^1 x 2 on the interval[ 0 , 1 ]is

equal to 0. Ifg(x)is identically 0, choosex 0 to be any number between 0 and 1. Otherwise,