Conditions Needed for Interchange of Double Limits

[disregardthat]

What conditions must f satisfy if

$$\lim_{x \to a} \lim_{y \to b} f(x,y)=\lim_{y \to b} \lim_{x \to a} f(x,y)$$

where $$\lim_{x \to a} f(x,y)$$ and $$\lim_{y \to b} f(x,y)$$ exists and are finite?

 

[disregardthat]

No one?

 

[thofer]

Without condition you're statement is not true, consider
$$\frac{y^2}{x^2+y^2}.$$

 

[uart]

What conditions must f satisfy if

$$\lim_{x \to a} \lim_{y \to b} f(x,y)=\lim_{y \to b} \lim_{x \to a} f(x,y)$$

where $$\lim_{x \to a} f(x,y)$$ and $$\lim_{y \to b} f(x,y)$$ exists and are finite?

Good question. I'm fairly sure that continuity in both x and y would be a sufficient condition.

 

[g_edgar]

f should be continuous at (a,b) ... that is, continuous as a function of two variables. Continuous separately in each of x and y is not enough.

 

[uart]

f should be continuous at (a,b) ... that is, continuous as a function of two variables. Continuous separately in each of x and y is not enough.

Arh yes, thanks for the clarification g_edgar. :)

BTW, what is the simplist definition of continuity in this case. I was thinking of something like :

[itex] \exists \, \, \epsilon > 0 \, : \, |f(x+dx,y+dy) \, - \, f(x,y) | \leq \, k \, ||(dx,dy)|| [/tex] whenever $||(dx,dy)|| \leq \epsilon$.

Is that basically correct?

 

[thofer]

If $$f$$ is continuous the we certainly have $$\lim_{x\rightarrow a} \lim_{y\rightarrow b} f(x,y) = \lim_{y\rightarrow b} \lim_{x\rightarrow a} f(x,y).$$
But you do not need that much. Consider
$$g(x,y) = \frac{xy}{x^2+y^2}.$$
The function is discontinuous at $$(0,0)$$, since $$\lim_{t\rightarrow 0} g(t,t) = 1/2 \neq 0 = \lim_{t\rightarrow 0} g(t,0)$$.
But we have $$\lim_{x\rightarrow 0} \lim_{y\rightarrow 0} g(x,y) = \lim_{y\rightarrow 0} \lim_{x\rightarrow 0} g(x,y)=0,$$ and $$\lim_{x\rightarrow 0} g(x,y) = 0 = \lim_{y\rightarrow 0} g(x,y)$$ exist and are finite.

 

[g_edgar]

Arh yes, thanks for the clarification g_edgar. :)

BTW, what is the simplist definition of continuity in this case. I was thinking of something like :

[itex] \exists \, \, \epsilon > 0 \, : \, |f(x+dx,y+dy) \, - \, f(x,y) | \leq \, k \, ||(dx,dy)|| [/tex] whenever $||(dx,dy)|| \leq \epsilon$.

Is that basically correct?

No that's not it. Try again!

 

[disregardthat]

Thanks for the replies.

Now, consider $$f(x,y)=x^y$$ on $$(0,\infty)$$ in both variables. Is the function is continuous in both variables on the interval, and not only seperately? We have existing limits as $$x$$ and $$y \to 0$$ independently. They are 0 and 1 respectively. However, the resulting limit depends on the order of the limit composition.

How do you account for this example? What conditions do f fail to satisfy? And what is the difference between continuity in two variables, and continuity in two variables seperately?

 

[g_edgar]

Thanks for the replies.

Now, consider $$f(x,y)=x^y$$ on $$(0,\infty)$$ in both variables. Is the function is continuous in both variables on the interval, and not only seperately?

Yes, $f$ is continuous on the whole product set $(0,\infty) \times (0,\infty)$.

We have existing limits as $$x$$ and $$y \to 0$$ independently. They are 0 and 1 respectively. However, the resulting limit depends on the order of the limit composition.

How do you account for this example? What conditions do f fail to satisfy? And what is the difference between continuity in two variables, and continuity in two variables seperately?

The condition that fails: $f$ is not continuous at the point $(0,0)$

Remember what I said back there? "f should be continuous at (a,b)" ?

 

[disregardthat]

How is continuity defined in a point for a function of 2 or more variables?

 

[Moo Of Doom]

A function f is continuous at a point v, if for every e>0 there is a d>0 such that
||f(v + u) - f(v)|| < e whenever ||u|| < d.