Results 1 to 2 of 2

Math Help - Differentiable Proof 2

  1. #1
    Member thaopanda's Avatar
    Joined
    Sep 2009
    From
    Worcester, Massachusetts
    Posts
    85

    Differentiable Proof 2

    Suppose that f: R \rightarrow R is differentiable and that for every a,b \in R there holds f (a + b) = f(a) + f(b).

    Prove that f '(x) = f '(0) for all x \in R. What kind of function is f?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Nov 2009
    Posts
    13
    Start by noticing that f(0)=0. Indeed, putting a=b=0, one gets f(0)=2f(0), hence f(0)=0.
    Now put a=x, b=h:

    f(x+h)-f(x)=f(h)=f(h)-f(0)
    Divide by h to get
    (f(x+h)-f(x))/h=(f(h)-f(0))/h

    Take the limit h->0 to finally get
    f'(x)=f'(0).

    This means that f'(x)=a is a constant: f(x)=cx+d, but d must be zero in order to the first relation to hold.

    Hence f(x)=cx. Those are called linear functions.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. A proof about differentiable functions
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 9th 2011, 12:07 AM
  2. Replies: 9
    Last Post: December 17th 2010, 09:13 AM
  3. Replies: 1
    Last Post: May 4th 2010, 03:28 AM
  4. Proof that G is differentiable on \R & compute G'
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: April 20th 2010, 12:21 PM
  5. Differentiable Proof
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: November 1st 2009, 06:17 PM

Search Tags


/mathhelpforum @mathhelpforum