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Differentiability of a two variable function with parameter

  1. Nov 9, 2011 #1
    1. The problem statement, all variables and given/known data

    For wich parameter [tex]\alpha\in\mathbb{R}[/tex] the function:
    [tex]f(x, y)= \begin{cases}|x|^\alpha \sin(y),&\mbox{ if } x\ne 0;\\ 0, & \mbox{ if } x=0\end{cases}[/tex]

    is differentiable at the point (0, 0)?

    3. The attempt at a solution

    For α<0, the function is not continue at (0, 0), so it is not differentiable. I checked it :)

    For α≥0 i have troubles, many troubles. I need your help.

    I started evalueting the partial derivates using the definition:

    [tex]\partial_x f(0,0):= \lim_{h\to 0} \frac{|h|^\alpha\sin(0)-0}{h}=0 [/tex]
    [tex]\partial_y f(0,0):= \lim_{h\to 0} \frac{|0|^\alpha\sin(h)-0}{h}=0 [/tex]

    (I think i must split the cases α=0 and α>0, right? |0|^α has no sense if α=0, but if α=0 then f(x,y)= sin(y)... I'm confuse)

    Anyway, i need to find the value of the limit (if it exists):

    [tex]\lim_{(h,k)\to (0, 0)}\frac{f(h, k)}{\sqrt{h^2+k^2}}[/tex] and show that its value is 0 right? I have problems with two variables limits...

    Any helps will be appreciated...

    Please, if you see there are mistakes in English languange, correct me :)
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Nov 9, 2011 #2


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    See if the limit of each of the following exists as (x, y) → (0, 0):
    [itex]\displaystyle \frac{\partial f(x,\,y)}{\partial x}[/itex] and [itex]\displaystyle \frac{\partial f(x,\,y)}{\partial y}[/itex]​
    Last edited: Nov 9, 2011
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