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) (is not equal to zero)= 0 for all x such that λ(x) = q(x)(p(x) + 1).

Answer 1

We set $\lambda=sin(2\pi\,x)$, $x\neq n+\frac12$, $n\in{\mathbb{N}}$ . For $x=n+\frac12$ we put $\lambda=1$. Then we put $q(x)=1,$ $p(x)=sin(2\pi x)-1,$ $x\neq n+\frac12,n\in{\mathbb{N}}$ . For $x=n+\frac12$ we set $p(x)=1$.