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Is f(x,y) continuous?


*****

Problem:

Lef f be the function defined by

> ;

                                             2
                                          x y
                               f(x, y) = -------
                                          2    4
                                         x  + y

when (x,y) is not (0,0) and define

> ;

                                  f(0, 0) = 0

Is f continuous at (0,0)? Explain.

Solution:


If we use the path y=a x^m with x->0 we obtain,

> Limit('x*a^2*x^(2*m)/(x^2+a^4*x^(4*m))',x=0) =
> limit(a^2*x^(2*m-1)/(1+a^4*x^(4*m-2)),x=0);

                          2  (2 m)                2  (2 m - 1)
                       x a  x                    a  x
               lim    -------------- =  lim    -----------------
              x -> 0   2    4  (4 m)   x -> 0       4  (4 m - 2)
                      x  + a  x                1 + a  x

when m=1/2 this last limit produces

> ;

                                       2
                                      a
                                    ------
                                         4
                                    1 + a

so the limit depends on a and thus it DNE. The function f is therefore not continuous at (0,0) since it doesn't have a limit at (0,0).


Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Fri Mar 23 15:30:53 EST 2001