Question

You have given a function λ : R → R with the following properties (x ∈ R, n ∈ N):

λ(n) = 0 , λ(x + 1) = λ(x) , λ(n+1/2)=1
Find two functions p, q : R → R with q(x)6=0 for all x such that λ(x) = q(x)(p(x) + 1).

Answer 1

for $n \in \N : p(n) = -1, q(n) = c$ (can be any function, except 0, by condition)

$p(n+\frac{1}{2}) = 0, q(n+\frac{1}{2}) = 1$

for $x \in \R : p(x) = 0, q(x) = x \mod 1$

where $x \mod 1 = x \% 1 = x - \lfloor x \rfloor$in another words it is remainder of division by 1