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Homework Statement
Let F:R->R be a function such that, for all x,y belonging to R, we have F(x+y)=F(x)+F(y) and F(xy)=F(x)F(y). Prove that F is one of the following two functions:
i> f(x)=0
ii> f(x)=x
(Hint : At some point in your proof, the fact that every positive real number is the sqaure of a real number will be valuable)
Homework Equations
The Attempt at a Solution
Let f(x)=x^n. Then f(x+y)=f(x)+f(y) is not satisfied. So f(x) can't be a polynomial. Similarly f(x) can't contain e^x, logx, any trigonometric function.
Now let f(x)=ax, where a=!0,1.
then f(x+y)=f(x)+f(y)=2ax is satisfied but not f(xy)=f(x).f(y) as f(x^2)=a.x^2 but f(x).f(x)=a^2.x^2. NOw a = a^2 iff a= 0 or 1. hence the proof.
My confusion is (as I am new to prrof oriented maths.), I have actually never used the hint. So, did I skip any argument?
Thanks in advance.