f(x) is a continues function on (-infinity,+infinity) for which f(x+y)=f(x)+f(y) prove that there is parameter a for which f(x)=ax for every real x i was given a hint to solve it for x in Q there is no much thing i can do here for which i can use theorems the only thing i am given that its continues lim f(x)=f(x) ??
1. Prove by induction that f(nx)= nf(x) for any positive integer n and any real number x. 2. From that, taking x= 0, show that f(0)= 0. From here on, n will represent any integer and x any real number. 3. Prove, by looking at f((n+(-n))x), that f(-nx)= -f(nx). 3. Prove, by looking at f(n(x/n)), that f(x/n)= f(x)/n. 4. Prove that, for any rational number, r, f(r)= rf(1). 5. Use the continuity of f to show that f(x)= xf(1) for any real number, x.
f(kx + x) = f(kx) + f(x) = kf(x) + f(x) = (k + 1)f(x) The first step of the chain of equality above comes from the assumption in the original problem.