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 *y* = *x*^{2}, it has a limit of 0.5.
Since taking different paths toward the same point yields different values for the limit, the limit does not exist.

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 f_{y}(x):= f(x,y) are continuous in x (for any fixed y).
Similarly, all f_{x} 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.