Find two functions h, k: R rightarrow R such that neither h nor k is a constant map, but k o h is a
constant map.
Solution
Consider the following two functions:
h(x) = x - [x]
k(x) = [x]
Where [x] is the biggest integer less than or equal to x. It's obvious that h and k are not constant
functions.
We have:
x-1 < [x] <= x
Therefore:
0 <= x - [x] < 1
So for all values of x: 0 <= h(x) < 1
So we have:
koh = k(h(x)) = [h(x)]
and because 0 <= h(x) < 1 : [h(x)] = 0
Thus
k(h(x)) = [h(x)] = 0
Therefore koh is a constant function.