Largest subset in which a function is continuous

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Main Question or Discussion Point

Take for ex f(x,y) = x/y. Domain is all (x,y) except for y = 0. It's continuous everywhere except for y = 0. Is this always the case? The function is continuous everywhere in its domain?

No. E.g. take $f(x) = \begin{cases} 1 & \text{ if } x= 0\\ 0 & \text{ if } x\neq 0 \end{cases}$. It's defined for all real numbers, so its domain is $\mathbb{R}$ whereas it is not continuous at $x=0$. Of course you can easily find more complicated examples.