Proving that cross partials in R^2 are equal if

[GridironCPJ]

There is a theorem stating the following:

Let f be defined in a neighborhood of (x0, y0) in R^2. Suppose f has partial derivatives f_1, f_2, f_12, and f_21 in this neighborhood and that the cross partials f_12 and f_21 are continuous at (x0, y0). Then the cross partials f_12 and f_21 are equal at (x0, y0).

This is a theorem proven in TBB's Elementary Real Analysis. It is stated that there are other conditions that could be met instead. I will reqrite the theorem with this other condition:

Let f be defined in a neighborhood of (x0, y0) in R^2. Suppose f has partial derivatives f_1, f_2, f_12, and f_21 in this neighborhood and that the partials f_1 and f_2 are differentiable at (x0, y0). Then the cross partials f_12 and f_21 are equal at (x0, y0).

How would you prove this or start a proof of this? No, this is not a homework problem, this is a theorem I'm curious about myself for my own satisfaction. Also, if my notation is not clear, f_1 is a partial derivative with respect to x, f_2 is the partial derivative with respect to y.

 

[lugita15]

Well, the result you want to prove follows trivially from the theorem in your book, along with the theorem that differentiability implies continuity. So I assume the latter is what you want to prove, right? See here.

 

[GridironCPJ]

Well, the result you want to prove follows trivially from the theorem in your book, along with the theorem that differentiability implies continuity. So I assume the latter is what you want to prove, right? See here.

Yes, differentiability implies continuity, but we are given f_1 and f_2 as being differentiable at (x0, y0), so this tells us that f_1 and f_2 are continuous there, but this does not say anything about the continuity of f_12 and f_21, which is the criteria in the first theorem I gave (the one proven in TBB).

 

[gopher_p]

You might try using the fact that derivatives have the intermediate value property.

Or just look at the proof of the original statement and see where and how the continuity of $f_{12}$ and $f_{21}$ is used. You might see how that condition is a bit more than necessary.