Mean-value theorem for functions of two real variables?

[Hodgey8806]

I am studying a problem that want's me to show that differentiable function at z0 is continuous at z0 = x0 + iy0.
It hints to use the mean-value theorem for functions of two real variables. But I can't find any true applicable one on the web. I searched the book with no theorem so named either. May I see the form it takes compared to the mean-value theorem of single variables? Thanks!

 

[Office_Shredder]

The only thing I can think is that if you consider a path connecting two points, $$\gamma(t)$$, and a two dimensional function $$f(x,y)$$ you can apply the mean value theorem to $$f(\gamma(t))$$

 

[Hodgey8806]

A two dimensional problem would be a good version if you were suggesting to different versions. Is there a theorem of the form similar to mean-value theorem for single variable functions?

 

[Office_Shredder]

A two dimensional problem would be a good version if you were suggesting to different versions.

I don't understand what you mean by this

 

[Hodgey8806]

Sorry, I'm not sure how to word it exactly. I'm really lost on how to use the mean value theorem for a two-variable functions.

I could give you the problem if you'd rather see it. Thanks for your help

 

[JG89]

Does your function map (x,y) to a real number? If so, here is a useful theorem for you:

Let A be open in R^m. Let f: A -> R be a differentiable function on A. If A contains the line segment with end points a and a + h, then there is a point c = a + th with 0 < t < 1 of this line segment such that f(a + h) - f(a) = Df(c) * h

 

[Hodgey8806]

I appreciate the help, but I'm not sure how to really apply it to this problem.

I hope I'm not breaking the rules of this forum with this piece, but the problem says:

Let f(z) = u(x,y) + iv(x,y) be differentiable at z0. Show that u and v are continuous at z0 = x0 + iy0. Hint: Use the mean-value theorem for functions of two real variables.

I don't exactly know what it is looking for me to do in this problem. Thanks!