Stuck on proof regarding partial derivatives

  • Thread starter Yami
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Homework Statement


Suppose the function f:R^2→R has 1st order partial derivatives and that
δf(x,y)/δx = δf(x,y)/δy = 0 for all (x,y) in R^2.
Prove that f is constant; there exists c such that f(x,y) = c for all (x,y) in R.

There's a hint as well:
First show that the restriction of f:R^2→R to a line parallel to one of the coordinate axes is constant.

Homework Equations


The chapter defines partial derivatives in general:
Let O be open in R^n and i be an index with 1≤i≤n. A function f:O→R has a partial derivative with respect to its ith component at the point x in O provided that
lim_(t→0)(f(x + t*e_i) - f(x))/t exists.


The Attempt at a Solution


The chapter has no mention of integrals but I tried using integrals anyway since I have no idea what to do with the given hint.

0 =
y
∫δf(x,t)/δy dt +
0
x
∫δf(t,0)/δy dt
0
= f(x,y) - f(x,0) + f(x,0) - f(0,0)
= f(x,y) - f(0,0)

implying f(x,y) = f(0,0) for all (x,y).
But then I realized that the partial derivatives being 0 means their antiderivatives are constant, so that wont work out.
Now I'm looking at the hint again but, I'm really stumped on how to use it. The chapter mentions the Mean Value Theorem. I thought of trying that somehow, but it's not given that the function is continuous. Can anyone point me in the right direction?
 

Answers and Replies

  • #2
LCKurtz
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Set y = c, constant, and consider h(x) = f(x,c). Can you apply the MVT to that?
 
  • #3
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Well, another reason I'm having trouble with this problem is I'm having trouble believing it's true.
What if f(x,y) =
{2 if (x,y) = (0,0),
{3 if (x,y) ≠ (0,0).

Wouldn't the partial derivatives be 0 still? Again it's not given that the function is continuous. Do I have to assume it's continuous?
 
  • #4
LCKurtz
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Well, another reason I'm having trouble with this problem is I'm having trouble believing it's true.
What if f(x,y) =
{2 if (x,y) = (0,0),
{3 if (x,y) ≠ (0,0).

Wouldn't the partial derivatives be 0 still? Again it's not given that the function is continuous. Do I have to assume it's continuous?

If the partial derivative with respect to x exists at (x,c) then the function of x given by f(x,c) must be continuous. Same for f(c,y). Assuming the derivative exists implies continuity.
 
  • #5
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Ah, yes. But this section in the book specifically states that
"For n > 1, a function f: R^n→R that has first order partial derivatives need not be continuous," and even provides an example:
f(x,y) =
{xy/(x^2+y^2) if (x,y) ≠ (0,0)
{0 if (x,y) = (0,0)

The partials at (0,0) are 0.

Okay, I'm going try this:
Suppose f is not continuous. Then f is not constant, since f being constant implies it's continuous. Then we are done?

Then with that assertion that f should be continuous I can use MVT. Yes?
 
  • #6
LCKurtz
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If the partial derivative with respect to x exists at (x,c) then the function of x given by f(x,c) must be continuous. Same for f(c,y). Assuming the derivative exists implies continuity.

Ah, yes. But this section in the book specifically states that
"For n > 1, a function f: R^n→R that has first order partial derivatives need not be continuous," and even provides an example:
f(x,y) =
{xy/(x^2+y^2) if (x,y) ≠ (0,0)
{0 if (x,y) = (0,0)

The partials at (0,0) are 0.

Okay, I'm going try this:
Suppose f is not continuous. Then f is not constant, since f being constant implies it's continuous. Then we are done?

Then with that assertion that f should be continuous I can use MVT. Yes?

But I didn't say the function f(x,y) was continuous as a function of two variables. What you do have is that as a function of one variable with the other held constant it is continuous. And its derivative is given to exist and is 0.
 
  • #7
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Oh, I think I see now. For a function of one variable, differentiability implies continuity. So that's where the hint fits in? Use the line parallel to an axis to equate it to a function with an interval domain to imply continuity?

So I can do something like this:
I = {(x,c) | c = some constant}, the line.
g:I→R ; g(x) = f(x,c). Since g is differentiable on I it is continuous?
 
  • #8
LCKurtz
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Oh, I think I see now. For a function of one variable, differentiability implies continuity. So that's where the hint fits in? Use the line parallel to an axis to equate it to a function with an interval domain to imply continuity?

So I can do something like this:
I = {(x,c) | c = some constant}, the line.
g:I→R ; g(x) = f(x,c). Since g is differentiable on I it is continuous?

You have a function of one variable whose derivative is 0 and you are trying to show it is a constant. Perhaps if you show it is constant along the line where y = c that would get you started. Look in your calculus book to see how f'(x) identically 0 implies f(x) is a constant. I bet you will see a reference to the MVT.
 
  • #9
LCKurtz
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And I might add that once you have f(x,y) is constant along lines (x,c) and (a,y) you need to show that f is the same constant everywhere.
 
  • #10
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I think I've written out a satisfying proof. Thanks for your help.
 

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