Results 1 to 2 of 2

Thread: Proving continuity of function

  1. #1
    Member
    Joined
    Aug 2009
    Posts
    130

    Proving continuity of function

    I would just like someone to check this proof for me. I am just learning this material so I would like to make sure I am doing the proofs properly.

    Question: Show that if $\displaystyle A \subset R^n $ and $\displaystyle f: A \rightarrow R $ and if the partial derivatives $\displaystyle \partial f_{x_i} $ exist and are bounded in a neighborhood of $\displaystyle \vec{a} \in R^n $, then f is continuous at $\displaystyle \vec{a} $


    Proof: We have to show that $\displaystyle |f(\vec{a} + t\vec{e_i}) - f(\vec{a})| \rightarrow 0 $ as $\displaystyle t \rightarrow 0 $ where $\displaystyle t $ is a real number and $\displaystyle \vec{e_i} $ is any basis vector for $\displaystyle R^n $.

    Let $\displaystyle M $ be an upper bound for all the partial derivatives of f, in some neighborhood of $\displaystyle \vec{a} $. Observe that $\displaystyle 0 \le \lim_{t \rightarrow 0} |f(\vec{a} + t\vec{e_i}) - f(\vec{a})| = \lim_{t \rightarrow 0} |t| |\frac{f(\vec{a} + t\vec{e_i}) - f(\vec{a})}{t}| = \lim_{t \rightarrow 0} |t| \lim_{t \rightarrow 0} |\frac{|f(\vec{a} + t\vec{e_i}) - f(\vec{a})}{t}| $ $\displaystyle = \lim_{t \rightarrow 0} |t| \partial f_{x_i} \le \lim_{t \rightarrow 0} |t| M = 0 $ and so $\displaystyle |f(\vec{a} + t\vec{e_i}) - f(\vec{a})| \rightarrow 0 $ as desired. QED
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Feb 2010
    Posts
    422
    Did you use the fact that the derivatives exist and are bounded in a neighborhood of $\displaystyle a$?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proving continuity
    Posted in the Differential Geometry Forum
    Replies: 12
    Last Post: May 15th 2011, 11:48 AM
  2. Proving Continuity
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Mar 20th 2011, 10:49 PM
  3. proving continuity
    Posted in the Differential Geometry Forum
    Replies: 16
    Last Post: Dec 3rd 2010, 05:09 PM
  4. Proving Continuity of a function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Mar 1st 2009, 09:51 AM
  5. Proving Continuity
    Posted in the Calculus Forum
    Replies: 5
    Last Post: May 19th 2008, 04:27 AM

Search Tags


/mathhelpforum @mathhelpforum