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Proving partial deviatives not continous

  1. Nov 24, 2012 #1
    1. The problem statement, all variables and given/known data
    f(x,y) = y^2 + (x^3)*sin(1/x) when x =/= 0
    = y^2 when x = 0

    i want to prove fx(x,y) is not continous at (0,0)
    2. Relevant equations



    3. The attempt at a solution
    i found when x=/=0 , fx = 3(x^2)sin(1/x) - xcos(1/x) -----eq(1)
    and limit(x,y -> 0,0) eq(1) = 0 as sin and cos is bounded
    and the actual fx(0,0) = limit(h->0) (f(h,0)-f(0,0))/h = lim(h->0) (h^2)*sin(1/h) = 0
    it seem limfx(0,0) = fx(0,0)
    so i cannot conclude that fx is not continous at (0,0)
    where did i go wrong?
     
  2. jcsd
  3. Nov 24, 2012 #2
    can someone offer help plz?.
     
  4. Nov 24, 2012 #3
    I think the partial derivative is actually continious. Graphing it shows it has no oscillation, which is a quantitative definition of continuity.
     
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