  The function f:X -> Y is called k-monotonically increasing if there is a
partition X = X_1 U ... U X_k such that f|X_i : X_i -> Y is monotonically
increasing for i=1,...,k. It is proved that a one-to-one function f:N -> N is
k-monotonically increasing if and only if every set of k+1 positive integers
contains two integers x,x' with x < x' such that f(x) <= f(x'). The function
f:X \to Y is called k-monotonic if there is a partition X = X_1 U ... U X_k
such that f|X_i : X_i -> Y is monotonically increasing or monotonically
decreasing for i=1,...,k. It is also proved that there does not exist a
k-monotonic function from N onto Q.
