Is There a Parameter a Such That f(x) Equals ax for All x?

[transgalactic]

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)

??

 

[HallsofIvy]

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.

 

[transgalactic]

regarding 1:
f(kx)=kf(x) given
prove f(kx + x)=(k+1)f(x)

i don't know how to use the given
??

 

[Mark44]

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.