Results 1 to 2 of 2

Math Help - 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  A \subset R^n and  f: A \rightarrow R and if the partial derivatives  \partial f_{x_i} exist and are bounded in a neighborhood of  \vec{a} \in R^n , then f is continuous at  \vec{a}


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

    Let  M be an upper bound for all the partial derivatives of f, in some neighborhood of  \vec{a} . Observe that  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}|  = \lim_{t \rightarrow 0} |t| \partial f_{x_i} \le \lim_{t \rightarrow 0} |t| M = 0 and so  |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 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, 12:48 PM
  2. Proving Continuity
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 20th 2011, 11:49 PM
  3. proving continuity
    Posted in the Differential Geometry Forum
    Replies: 16
    Last Post: December 3rd 2010, 06:09 PM
  4. Proving Continuity of a function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 1st 2009, 10:51 AM
  5. Proving Continuity
    Posted in the Calculus Forum
    Replies: 5
    Last Post: May 19th 2008, 05:27 AM

Search Tags


/mathhelpforum @mathhelpforum