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Continuity and differentiability in two variables

  1. Nov 24, 2014 #1
    Hi

    If the function ##f(x,y)## is independently continuous in ##x## and ##y##, i.e.
    [itex]f(x+d_x,y) = f(x,y) + \Delta_xd_x + O(d_x^2)[/itex] and [itex]f(x,y+d_y) = f(x,y) + \Delta_yd_y + O(d_y^2)[/itex]
    for some finite ##\Delta_x##, ##\Delta_y##, and small ##\delta_x##, ##\delta_x##,

    does it mean that it is continuous in both?
    [itex]f(x+d_x,y+d_y) = f(x,y) + \Delta_xd_x +\Delta_yd_y+O(d_x^2,d_y^2)[/itex]

    How about differentiability? (if the function is independently differentiable in ##x## and ##y##, is it differentiable in both ##x## and ##y##?)

    Cheers
    wavingerwin
     
  2. jcsd
  3. Nov 27, 2014 #2

    Stephen Tashi

    User Avatar
    Science Advisor


    By "is continuous in both", do you just mean "is continuous"? Then no.. See Ch 9, section 1 of Counterexamples In Analysis (p 115 of the book, p 140 of the PDF) http://www.kryakin.org/am2/_Olmsted.pdf
     
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