continuity

[shizzy]

The question:

Let f(x,y) =            x^2
                   --------------
                    x^2 + y^2

Is it possible to define f(0,0) so that f will be continuous at (0,0)??

The book says no. I don't understand why exactly. Is it because although you can take a limit at a point, that doesn't mean it's continuous? I think I read that the only way it's continuous at a point is if the limit and the function value at that point agree. So here the limit value is..1 and the function value is...undefined?? Or??? Here is possibly the source of my confusion. Some work I have done: @ = theta

Let x = rcos(@) and Let y = rsin(@)
x^2 = (r^2)cos^2(@) [[squaring x here]]
y^2 = (r^2)sin^2(@) [[squaring y here]]

now setting equations equal:

x^2 + y^2 = (r^2)cos^2(@) + (r^2)sin^2(@)
x^2 + y^2 = (r^2)(cos^2(@) + sin^2(@)) [[[factor out an r^2]]]
x^2 + y^2 = r^2(1)
x^2 + y^2 = r^2

So using this on the original equation:

      x^2                   (r^2)cos^2(@)
  ---------------  =      ----------------  = cos^2(@)
    x^2 + y^2                   r^2

the trig method was given in the book, converting to polar coordinates it said. I remember that vaguely(polar stuff that is)....

I feel like I'm missing something here. It seems WAY too easy to be able to change that equation of 2 variables into a simple cos squared function. Any insight on this would be great. Also if anyone has tips on how to format my math equation so they are more readable, that would be helpful to me also. Thanks!

 

[pka]

Because the question is a limit in ℜ<SUP>2</SUP>, we can consider different paths.
The limit on the path y=x is ½.
But the limit on the path y=0 is 1.
So can we make this function continuous at (0,0)?

 

[shizzy]

Okay so the limit definately doesn't exist then. I'm not 100% on the continuity part though. I *think* it's not continuous since there is no limit. Looks like I need to do a little more research on continuity. I'll be back! Thanks!

 

[shizzy]

okay, thanks for the help, think I have it:

Shortening a definition, a function f(x,y) is continuous at a point if the function value at that point, and the limit are equal.

Also, a theorem says that a function is continuous everywhere except where there is division by zero (in this case).

So by the definition this function is not continuous at the point, and by the theorem since it's not a removable discontinuity, it is not continuous at the point.