Question

E(x )=[2x]-[x]-[x+1/2],
E[x+1/2] =E(X), oricare ar fi x∈R

Answer 1

Avem urmatoarea proprietate:

$[x+n]=[x]+ n\ , \ \ \forall \ n\in Z$

[tex]E(x+\frac{1}{2})=[2(x+\frac{1}{2})]-[x+\frac{1}{2}]-[x+\frac{1}{2}+\frac{1}{2}]=\\\\ =[2x+1] -[x+\frac{1}{2}]-[x+1]=[2x]+1-([x]+1)-[x+\frac{1}{2}]=\\\\ =[2x]-x-[x+\frac{1}{2}]=E(x)[/tex]