A study of limits and continuity in multivariable calculus yields many counter-intuitive results not demonstrated by single-variable functions .
For example, there are scalar functions of two variables with points in their domain which give a particular limit when approached along any arbitrary line, yet give a different limit when approached along a parabola.
For example, the function
Continuity in each argument does not imply multivariate continuity: For instance, in the case of a real-valued function with two real-valued parameters, f(x,y), continuity of f in x for fixed y and continuity of f in y for fixed x does not imply continuity of f. As an example, consider
It is easy to check that all real-valued functions (with one real-valued argument) that are given by fy(x):= f(x,y) are continuous in x (for any fixed y). Similarly, all fx are continuous as f is symmetric with regards to x and y. However, f itself is not continuous as can be seen by considering the squence f(1/n,1/n) (for natural n) which should converge to f(0,0)=0 if f was continuous. However, lim f(1/n,1/n) = 1.