MathDB

It is given function

f : A \rightarrow \mathbb{R}

(A\subseteq \mathbb{R})

such that

f(x+y)=f(x)\cdot f(y)-f(xy)+1; (\forall x,y \in A)

f : A \rightarrow \mathbb{R}

(\mathbb{N} \subseteq A\subseteq \mathbb{R})

is solution of given functional equation, prove that:

f(n)=\begin{cases} \frac{c^{n+1}-1}{c-1} \text{, } \forall n \in \mathbb{N}, c \neq 1 \\ n+1 \text{, } \forall n \in \mathbb{N}, c = 1 \end{cases}

where

c=f(1)-1

a)

Solve given functional equation for

A=\mathbb{N}

b)

With

A=\mathbb{Q}

, find all functions

f

which are solutions of the given functional equation and also

f(1997) \neq f(1998)

Problem Statement