# 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>