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November 14th, 2011, 12:40 PM   #1
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differentiable function?

Let be a function on



and when .

Is differentiable at ? Is the differential of function is continuous at this point?

Is this function differentiable at ?

Can anyone give me a hand?
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November 14th, 2011, 01:37 PM   #2
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Re: differentiable function?

Quote:
Originally Posted by Camille91
Let be a function on



and when .

Is differentiable at ? Is the differential of function is continuous at this point?

Is this function differentiable at ?

Can anyone give me a hand?
You should clarify what you mean by differentiable - y only or both x and y.

For x = 1 and y near 0, f(x,y) ? 1/(2?y), which is not differentiable.
For x and y near 0, f(x,y) ? {?(x/y)}/2
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November 17th, 2011, 09:22 AM   #3
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Re: differentiable function?

Mathman, "differentiable" is defined for functions of two variables, separately from the existance of the partial derivatives. It can be shown, however, that if a function is differentiable at a point, its partial derivatives must be continuous in a neighborhood of that point.
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