Re: derivative of integral - textbook proof

On Tue, 24 Apr 2007 14:41:51 -0400, "G. A. Edgar"
<edgar@xxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

In article <240420071135374405%edgar@xxxxxxxxxxxxxxxxxxxxxxxxxxx>, G.
A. Edgar <edgar@xxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

Folland, ADVANCED CALCULUS, Theorem 4.47

A proof seems to use (something more general than):
If f(x,y) continuous on (0,1) x [0,1] and for each fixed y,
f(x,y) is continuously differentiable as a function of x,
then \partial f/\partial x is continuous on (0,1) x [0,1].

Is that correct?

OK here is a counterexample, f(x,y) = xy^3/(x^2+y^4)
this is continuous everywhere, in particular on (-1,1) x [0,1],
but the partial derivative f_x has f_x(0,y) = 1/y for y not 0,
so certainly f_x is not continuous at (0,0).

But your function f is not continuous at (0,0), so fails to satisfy
the hypothesis.

For example, approach (0,0) along the curve y=x^(1/4)

quasi
.

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