Results 1 to 6 of 6

Math Help - Differentiability Question

  1. #1
    Newbie
    Joined
    Oct 2009
    Posts
    15

    Differentiability Question



    I don't understand how to prove this considering f(x) is not specifically given....
    Last edited by JohnLeee; October 31st 2009 at 11:30 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943
    Quote Originally Posted by JohnLeee View Post


    I don't understand how to prove this considering f(x) is not specifically given....
    At zero, f(0)\leq ||0||^a\implies f(0)=0. So \frac{|f(x)-f(0)|}{||x-0||}\leq||x||^{a-1}.

    Take the limit of both sides as x\to0 and you get f'(0)=0.

    If a=1, all you know is that |f'(0)|\leq1.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Remember that for a function to be diff. at 0 there exists a linear transformation D_f(0) such that \vert \frac{f(h)-f(0)-D_f(0)h}{ \Vert h \Vert } \vert \rightarrow 0 as h \rightarrow 0. Take D_f(0) \equiv 0 then \vert \frac{f(h)-f(0)}{ \Vert h \Vert } \vert \leq \frac{ \Vert h \Vert ^a}{ \Vert h \Vert } = \Vert h \Vert ^{a-1} and since a-1>0 we have that this goes to 0 as h goes to 0. This proves that f is diff. at 0 and D_f(0) \equiv 0. When a=1 we can only bound by 1, but I can't think of a counter-example at the moment for this case.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943
    A good counterexample is f:\mathbb{R}\longrightarrow\mathbb{R} with f(x)=x.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Quote Originally Posted by redsoxfan325 View Post
    A good counterexample is f:\mathbb{R}\longrightarrow\mathbb{R} with f(x)=x.
    But f is differentiable at 0, I think we need a function such that \vert f(x) \vert \leq \Vert x \Vert and it's not diff. at 0 (If such a counter-example exists)
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943
    Oh, in that case how about x\sin(1/x)?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Continuity/Differentiability Question
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: June 13th 2011, 08:42 PM
  2. Differentiability Question?
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: January 27th 2011, 12:36 PM
  3. differentiability question
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: November 15th 2009, 11:13 AM
  4. Question about differentiability
    Posted in the Calculus Forum
    Replies: 4
    Last Post: May 9th 2009, 03:46 PM
  5. differentiability and continuity question
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 15th 2008, 12:42 PM

Search Tags


/mathhelpforum @mathhelpforum