I have recently started going into limits and continuity of multivariable functions and I was just wondering if there is an easy way to determine the continuity of various functions (e.g. polynomial, exponential, rational) in $\mathbb{R}^{2}$?
For instance, $$f(x, y) = x+y^{2},$$ $$f(x, y) = e^{x+y},$$ $$f(x, y) = \frac{x+y}{x-y}$$
Are all these functions continuous over $\mathbb{R}^{2}$? How do one determine so?
My hunch is that since the first equation is a polynomial function of two variables, it is continuous. The second equation is a composition of the exponential function with a polynomial function of two variables, hence it is also continuous. The last one is a rational function, so rather than being continuous over $\mathbb{R}^{2}$, it is continuous over the domain?
Can anyone confirm this for me?