Functional equation
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Functional equation
Prove that the only function \(\displaystyle{f:\mathbb{R}\longrightarrow \mathbb{R}}\) having the proprties
\(\displaystyle{i)\,\,f(x+y)=f(x)+f(y)\,\,\forall\,x\,,y\in\mathbb{R}}\)
\(\displaystyle{ii)\,\,f(x\,y)=f(x)\,f(y)\,\,\forall\,x\,,y\in\mathbb{R}}\)
\(\displaystyle{iii)}\) \(\displaystyle{\,\,f}\) is one to one
is the function \(\displaystyle{f=Id_{\mathbb{R}}}\) .
\(\displaystyle{i)\,\,f(x+y)=f(x)+f(y)\,\,\forall\,x\,,y\in\mathbb{R}}\)
\(\displaystyle{ii)\,\,f(x\,y)=f(x)\,f(y)\,\,\forall\,x\,,y\in\mathbb{R}}\)
\(\displaystyle{iii)}\) \(\displaystyle{\,\,f}\) is one to one
is the function \(\displaystyle{f=Id_{\mathbb{R}}}\) .
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