MathDB

Let

Z

and

R

denote the sets of integers and real numbers, respectively. Let

f: Z \rightarrow R

be a function satisfying: (i)

f(n) \ge 0

for all

n \in Z

(ii) f(mn)\equal{}f(m)f(n) for all

m,n \in Z

(iii) f(m\plus{}n) \le max(f(m),f(n)) for all

m,n \in Z

(a) Prove that

f(n) \le 1

for all

n \in Z

(b) Find a function

f: Z \rightarrow R

satisfying (i), (ii),(iii) and

0<f(2)<1

and f(2007) \equal{} 1

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