In my exams, the questions on continuity of multivariable functions are framed like "Discuss the continuity of $f(x,y)$ at $(a,b)$…" or "Is $f$ continuous at $(a,b)$…?", and likewise. If I know beforehand that the given function is discontinuous at a given point $(a,b)$, then I just need to find out two paths where the the value of $\lim_{(x,y)\to (a,b)} f(x,y)$ are different or not equal to $f(a,b)$. On the other hand, if I know that the function is continuous at the given point, then I can use the $\varepsilon-\delta$ definition of continuity to prove continuity. But since the questions don't seem to be giving much away about the continuity of the function at the given point, I'm not sure which approach should I take first while trying to solve the question.
Is there any quick and intuitive way to figure out whether a given multivariable function is continuous or not? At least in the cases where the given function is of the form $\frac{p(x,y)}{q(x,y)}$, where $p$ and $q$ are polynomials in $x$ and $y$? (Exceptions are fine. Just a generic and practically useful trick would do.)
For example, the function
$$f(x,y) =\begin{cases} \frac{x^{4}-y^{4}}{x^{4}+y^{4}} & (x,y)\neq (0,0) \\ 0 & (x,y)=(0,0) . \end{cases}$$
is discontinuous at $(0,0)$, while the function
$$g(x,y) =\begin{cases} \frac{x^{2}y^{2}}{x^{2}+y^{2}} & (x,y)\neq (0,0) \\ 0 & (x,y)=(0,0) . \end{cases}$$
is continuous at $(0,0)$. Is there any easy way to pick this just by looking at the functions?
The same issue exists with finding a limit and proving the existence of the limit.
Best Answer
Since you asked for heuristics: the degree being higher in the numerator than the denominator suggests continuity; being equal or less suggests discontinuity. (This is a heuristic and not a theorem, there are trivial counter-examples like $x/x$.)