Results 1 to 6 of 6

Math Help - Proving differentiability

  1. #1
    Senior Member Pinkk's Avatar
    Joined
    Mar 2009
    From
    Uptown Manhattan, NY, USA
    Posts
    419

    Proving differentiability

    Determine whether or not f(x,y)=x^{2}+3xy is differentiable.

    Now, from what I understand, in order to show that a function of two variables is differentiable at some point (a,b), I have to show that \frac{\partial f}{\partial x} and \frac{\partial f}{\partial y} are continuous at (a,b) and that f(x,y) itself is continuous at (a,b). I know that the partials are continuous for all (x,y), but how do I show that f(x,y) is continuous? (The problem does not ask if the function is differentiable at a specific point (a,b).) Thanks in advance.

    Same question with the equation f(x,y)=xy^{2}.
    Last edited by Pinkk; March 21st 2009 at 05:38 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    I think all they want is that, since x^2, x and y are all differentiable, then their product and sum is differentiable.
    This is not true in regards to division, like 1/x.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member Pinkk's Avatar
    Joined
    Mar 2009
    From
    Uptown Manhattan, NY, USA
    Posts
    419
    So since there are no points where either of those equations are undefined, the equations are continuous?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    All three (x,y, x squared)are continuous everywhere
    1/x is continuous everywhere except at zero, where it's not even defined.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,326
    Thanks
    1298
    Quote Originally Posted by Pinkk View Post
    So since there are no points where either of those equations are undefined, the equations are continuous?
    All polynomials have the property that where they are defined they are continuous (and, in fact, infinitely differentiable). That is, of course, not true for all functions.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member Pinkk's Avatar
    Joined
    Mar 2009
    From
    Uptown Manhattan, NY, USA
    Posts
    419
    Ah, okay. I wanted to make sure that was true for functions of two variables in three dimensions.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proving a claim regarding differentiability
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 19th 2011, 01:07 PM
  2. differentiability at x=0
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 8th 2011, 07:39 PM
  3. Replies: 1
    Last Post: November 25th 2010, 03:33 PM
  4. Proving differentiability
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 2nd 2010, 02:16 PM
  5. proving differentiability
    Posted in the Calculus Forum
    Replies: 7
    Last Post: August 4th 2007, 06:52 PM

Search Tags


/mathhelpforum @mathhelpforum