real analysis - I want to show that $f(x)=x.f(1)$ where $f:Rto R$ is additive.

I know that if $f$ is continuous at one point then it is continuous at every point. From this i want to show that $f(x)=xf(1).$ Can anybody help me to proving this?

Answer

HINTS:

1. 
   
   

   Look at $0$ first: $f(0)=f(0+0)=f(0)+f(0)$, so $f(0)=0=0\cdot f(1)$.

   
   


2. 
   

   Use induction to prove that $f(n)=nf(1)$ for every positive integer $n$, and use $f(0)=0$ to show that $f(n)=nf(1)$ for every negative integer as well.

   
   


3. 
   

   $f(1)=f\left(\frac12+\frac12\right)=f\left(\frac13+\frac13+\frac13\right)=\dots\;$.

   
   


4. 
   

   Once you’ve got it for $f\left(\frac1n\right)$, use the idea of (2) to get it for all rationals.

   
   
   


5. 
   

   Then use continuity at a point.