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

Answer 1

$$q(x)\not=0, x\in \R$$

Then for $n\in \N$

$$\lambda(n)=q(n)(p(n)+1)=0=>$$

$$=>p(n)+1=0=>p(n)=-1$$

Let $q(x)=1,x\in \R.$

Let $p(x)=|\sin(\pi x)|-1$

Then

$$\lambda(x)=1\cdot(|\sin(\pi x)|-1+1)$$

$$\lambda(x)=|\sin(\pi x)|$$

Check

$$\lambda(x+1)=|\sin(\pi(x+1))|=|-\sin(\pi x)|=$$

$$=|\sin(\pi x)|=\lambda(x), x\in\R$$

$$\lambda(n)=|\sin(\pi(n))|=0, n\in\N$$

$$\lambda(n+\dfrac{1}{2})=|\sin(\pi(n+\dfrac{1}{2} ))|=\sin(\dfrac{\pi}{2})=1, n\in\N$$

$$\lambda(x)=q(x)(p(x)+1)=1\cdot(|\sin(\pi x)|-1+1)$$