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

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Let x be a real number.The floor function \lfloor x \rfloor is defined to be the greatest integer less than or equal to the real number x. The fractional part function \{ x \}  is defined to be \{x\}= x -\lfloor x \rfloor. Define two functions p,q:\mathbb R\to\mathbb R with q(x) not equal to 0 for all x in the following way: p(x)=2\{x\}-1, q(x)=\begin{cases}4\{x\}-1,\ x\ne\frac{1}{4}+n,n\in\mathbb N \\2,\ x=\frac{1}{4}+n,n\in\mathbb N\end{cases}.

Then the function λ:\mathbb R\to\mathbb R, λ(x)= q(x)(p(x)+1)=\begin{cases}(4\{x\}-1)(2\{x\}), \ x\ne\frac{1}{4}+n,n\in\mathbb N\\ 1,\ \ \ x=\frac{1}{4}+n,n\in\mathbb N\end{cases} has the following properties:

\lambda(n)=(4\{n\}-1)(2\{n\}) =-1\cdot0=0,\ \ \lambda(x+1)= \lambda(x), and


for all x \in\mathbb R, n\in\mathbb N.

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Question ID: mtid-5-stid-8-sqid-3556-qpid-2255