How to show that $f : [0, \infty)^2 \setminus \{0,0\} \rightarrow [0, \infty)$ , $f(x,y) = x^y$ is not a continuous function?

Question

I've tried the path $x=y$ to obtain the limit 1 as $x \rightarrow 0+$ i.e $\lim_{x \rightarrow 0+} x^x = 1$ , but am unable to find another path for which the limit should apparently come different (if I understand it correctly) to prove that the function :

$f : [0, \infty)^2 \setminus \{0,0\} \rightarrow [0, \infty)$

$f(x,y) = x^y$

is not continuous.

Any help is appreciated.

Answer 1

$f(0,y)=0$ for all $y > 0$ and $f(x,0)=1$ for all $x>0$. Hence $\lim_{x\to 0,y \to 0} f(x,y)$ does not exist.