Results 1 to 4 of 4

Math Help - Proof of Differentiablity

  1. #1
    Newbie
    Joined
    Feb 2010
    Posts
    2

    Proof of Differentiablity

    f is differentiable on (a,b] and f(x)/(x-a) \to 1 as x \to a+, then f is unifomliy continuous on [a,b]

    Ok here is what I have so far. Let f(x) = x and g(x) = f(x)/(x-a) , g(a) = 0. Now I need to show that g is cont. at a then I will have g to be cont. on [a,b] and I will be done right? So how in the world do I go about showing g is cont. at a?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Apr 2008
    Posts
    1,092
    g is not continuous at a. You only need to show that f is continuous on [a, b].
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Feb 2010
    Posts
    2
    Got any tips on how I should show f is cont. on a since we are already given it's cont. on b
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by MichaelJordan View Post
    f is differentiable on (a,b] and f(x)/(x-a) \to 1 as x \to a+, then f is unifomliy continuous on [a,b]

    Ok here is what I have so far. Let f(x) = x and g(x) = f(x)/(x-a) , g(a) = 0. Now I need to show that g is cont. at a then I will have g to be cont. on [a,b] and I will be done right? So how in the world do I go about showing g is cont. at a?
    I don't understand, we can extend a,b]\mapsto\mathbb{R}" alt="fa,b]\mapsto\mathbb{R}" /> to \tilde{f}:[a,b]\mapsto\mathbb{R} by \tilde{f}(x)=\begin{cases} f(x) & \mbox{if} \quad x\ne a \\ 0 & \mbox{if} \quad x=a\end{cases}. It is clear that \tilde{f} is continuous for all points but a and \lim_{x\to a^+}f(x)=\lim_{x\to a^+}\frac{f(x)}{x-a}\cdot (x-a)=\lim_{x\to a^+}\frac{f(x)}{x-a}\cdot \lim_{x\to a^+}(x-a)=1\cdot 0=0=f(a). I feel as though I may have overlooked something smal lthough.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. proving differentiablity
    Posted in the Calculus Forum
    Replies: 4
    Last Post: January 12th 2012, 09:34 PM
  2. Differentiablity of sinx and cosx
    Posted in the Calculus Forum
    Replies: 8
    Last Post: December 10th 2010, 06:38 PM
  3. Replies: 2
    Last Post: October 21st 2009, 10:51 AM
  4. Show existence of Jacobian but non-differentiablity
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 23rd 2009, 05:06 AM
  5. Replies: 4
    Last Post: June 15th 2008, 04:55 PM

Search Tags


/mathhelpforum @mathhelpforum