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Continuously differentiable function

  1. Apr 7, 2014 #1
    1. The problem statement, all variables and given/known data

    Show that if ##f## is a continuously differentiable real valued function on an open interval in ##E^2## and ##\partial^2f/\partial x\partial y=0,## then there are continuously differentiable real-valued functions ##f_1,f_2## on open intervals in ##\mathbb{R}## such that ##f(x,y)=f_1(x)+f_2(y).##

    How can I prove this?

    2. Relevant equations

    None

    3. The attempt at a solution

    Let ##(x_0,y_0)\in E^2## and integrate twice:

    ##0=\int_{y_0}^y\int_{x_0}^x\partial_x(\partial_yf(x',y'))dx'dy'=\int_{y_0}^y(\partial_yf(x,y')-\partial_yf(x_0,y'))dy'=f(x,y)-f(x,y_0)-f(x_0,y)+f(x_0,y_0).##
     
  2. jcsd
  3. Apr 7, 2014 #2
    This looks fine. Was there any problem with this?
     
  4. Apr 7, 2014 #3
    Nope, I was just confirming. Thanks for confirming!
     
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