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[H.Quân- ĐHV]

find all continuous functions$ f$ from reals to it self such that

$(1+f(x)f(y) ) f(x+y) = f(x) + f(y) $.for all real numbers $ x,y$

(samasya , Vol.8 ,No.1 ,May 2001)

[H.Quân- ĐHV]

find all continuous functions$ f$ from reals to it self such that

$(1+f(x)f(y) ) f(x+y) = f(x) + f(y) $.for all real numbers $ x,y$

(samasya , Vol.8 ,No.1 ,May 2001)

Hint

- Cm$ |f(x)| \le 1 $
-$ f(x) = 1 $or $f(x) = -1$ thỏa mãn .
- $|f(x)| < 1$ , đặt $f(x) = tanhg(x)$ ( trong đó $tanh(x) = \dfrac{ e^x - e^{-x} }{e^x + e^{-x} } $ $f(x) = tanh(cx)$ ( $c$ bất kì)