MathDB

Morteza places a function

[0,1]\to [0,1]

(that is a function with domain [0,1] and values from [0,1]) in each cell of an

n \times n

board. Pavel wants to place a function

[0,1]\to [0,1]

to the left of each row and below each column (i.e. to place

2n

functions in total) so that the following condition holds for any cell in this board: If

h

is the function in this cell,

f

is the function below its column, and

g

is the function to the left of its row, then

h(x) = f(g(x))

for all

x \in [0, 1]

. Prove that Pavel can always fulfil his plan.

Problem Statement