How to solve this parameter proof

  1. 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. jcsd
  3. HallsofIvy

    HallsofIvy 40,504
    Staff Emeritus
    Science Advisor

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

    i dont know how to use the given
    ??
     
  5. Mark44

    Staff: Mentor

    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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