Question

Give an example of a function of two variables such that f(0,0) = 0 but f is NOT continuous at (0,0). Explain why the function f is NOT continuous at (0,0).

Answer 1

$f(x,y)=\begin{cases} \frac{2xy}{x^2+y^2} &(x,y) \neq (0,0) \\ 0 &(x,y)=(0,0) \end{cases}$

Choosing the path x = 0 we see that f (0, y) = 0, so

$\displaystyle\lim_{y→0} f (0, y) = 0$

Choosing the path x = y we see that

$f (x, x) = 2x^ 2 /2x^ 2 = 1$

so

$\displaystyle\lim_{x→0} f (x,x) = 1$

The Two-Path Theorem (if a function has two different limits along two different paths) implies that

$\displaystyle\lim_{(x,y)→(0,0)} f (x,y)$

does not exist.