Can someone please explain to me why this function is continuous? $$f(x,y)=x^{xy},\quad x>0, \quad y\in \mathbb{R} $$ I have thought like this that we can rewrite the question in this form: $f(x,y) = e^{xy\ln x}$ and $e^x$ is continuous and $\ln$ also continuous but I don't know how should I prove that x and y are continuous too?
I know the definition of continuity at a point "$a$" that for every $\epsilon>0$ there is $\delta>0$ such that $$ 0<|x-a|<\delta \implies |f(x,y)-f(a)|<\epsilon$$
Again, let me emphasize that I am giving you a very basic approach. The comments of others, if usable, give an easier solution.
Basic approach :
(1) Compute $|(x,y) - (a,b)|$ in terms of $x,y,a,b$.
(2) Compute $|f(x,y) - f(a,b)|$ in terms of $x,y,a,b$.
(3) Requiring that $|(x,y) - (a,b)| < \delta,$ form an inequality in $x,y,a,b$ and $\delta.$
(4) Requiring that $|f(x,y) - f(a,b)| < \epsilon,$ form an inequality in $x,y,a,b$ and $\epsilon.$
(5) Given any $\epsilon$ and given any $(a,b)$ in the domain of $f$, demonstrate that you can construct a $\delta$ such that whenever requirement (3) above is satisfied, requirement (4) will also be satisfied.
Note that with $(a,b)$ in the domain of $f$ requirements (3) and (4) above only pertain to any $(x,y)$ that satisfy the particular requirement and are also in the domain of $f$.