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February 16th, 2014, 08:30 PM   #1
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continuity and partial derivatives of a function

If and if given by

if and if

1. Values ??for is continuously throughout ?

2. Values ??for are the partial derivatives of throughout ?

3. Values ??for is differentiable at all ?
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February 21st, 2014, 01:11 PM   #2
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Re: continuity and partial derivatives of a function

1.) As the quotient of two differentiable functions, the given function is clearly continuous and differentiable everywhere but at where the denominator is zero. We therefore invoke some analytical tools to judge continuity at . is not continuous for . Going along the direction of , we get to approach zero in the limit as . However, going along the direction of , we get to approach .
is not continuous for either. Going along the direction of , we get to approach zero in the limit as . However going along the direction of , we get to approach in the limit as .
is continuous for . This is shown by using the binomial expansion of the expression in the numerator and observing that . Approaching in the function at the right hand side of the inequality, we get zero invariably, so that the function on the left hand side must also approach zero.

2.) Where is differentiable, its partial derivatives exist and are continuous and vice versa. Wherever are continuous then must be continuous also. Hence on ,


and since is not continuous at for and then are both not continuous. These partial derivatives are again not continuous for , as we see approaching in either case along the direction of and the direction of . The partial derivatives are continuous on for by using a similar argument of binomial expansion of terms in the numerator. Questions 2 and 3 have the same answer because continuity of the two partial derivatives is equivalent to differentiablity of the function .
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