
#1
Mar1309, 12:45 PM

P: 1,399

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) ?? 



#2
Mar1309, 01:35 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,900

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. 



#3
Mar1309, 02:54 PM

P: 1,399

regarding 1:
f(kx)=kf(x) given prove f(kx + x)=(k+1)f(x) i dont know how to use the given ?? 



#4
Mar1309, 02:59 PM

Mentor
P: 21,067

how to solve this parameter proof
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. 


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