Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How to solve this parameter proof

  1. Mar 13, 2009 #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. Mar 13, 2009 #2

    HallsofIvy

    User Avatar
    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. Mar 13, 2009 #3
    regarding 1:
    f(kx)=kf(x) given
    prove f(kx + x)=(k+1)f(x)

    i dont know how to use the given
    ??
     
  5. Mar 13, 2009 #4

    Mark44

    User Avatar
    Insights Author

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?