Possible Duplicate:
Proving that an additive function $f$ is continuous if it is continuous at a single point
Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions…)
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?
HINTS:
Look at $0$ first: $f(0)=f(0+0)=f(0)+f(0)$, so $f(0)=0=0\cdot f(1)$.
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.
$f(1)=f\left(\frac12+\frac12\right)=f\left(\frac13+\frac13+\frac13\right)=\dots\;$.
Once you’ve got it for $f\left(\frac1n\right)$, use the idea of (2) to get it for all rationals.
Then use continuity at a point.
