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If Partial derivatives exist and are continuos then function is differentiable

  1. Sep 25, 2011 #1
    1. The problem statement, all variables and given/known data
    Hi I'm just looking for a link to the proof of this theorem:
    if the partial derivatives of function f exist and are continuous at a point then the function is differentiable there

    Or even the name would be helpful
    Its not a homework assignment per say, just something that our lecturer said mentioned in passing but never gave a proof of and I would like it just for the sake of completeness :P

    2. Relevant equations

    Well I think the newton quotient is [ f(a+h,b+k) -(h)f(a,b+k)-(k)f(a+h,b) ]/Sqrt[h^2+k^2]
    (for 2 variables anyway)

    3. The attempt at a solution
    I'm guessing that it involves the mean value theorem but I'm not entirely sure :S
  2. jcsd
  3. Sep 25, 2011 #2


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    Science Advisor

    Any good Calculus text will have that proof- but I think it is too long and complicted to be given here.

    (By the way, the phrase is "per se"- "of itself".)
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