find all continous functions$$f(f(x+y))=f(x)+f(y)$$
My try:setting $y=0$ $$f(f(x))=f(x)+k$$ where $k=f(0)$
hence:let $f(x)=t$,$$f(t)=t+k$$ alternatively:$$f(x)=x+C$$
My question:1)Have i got all solutions?I think i am missing a generalised one.
2)does the word continous mean any thing important. I mean if they had not mentioned will there be more functions?