4. Sets, limits and continuity in multi-dimensional space

Sets in \mathbb{R}^n

A set X is bounded, if it is contained in a ball of some radius, i.e. if there exists r, such that for any x\in X, \|x\|<r.

A set is open if for every its point you can find a (small) radius such that the whole ball with centre in the point and this radius in contained in the set.

A set is closed, if its complement (everything outside of it) is an open set.

A set is compact if after covering it with (maybe infinitely many) open balls, you can choose only finitely many of them which still cover it. From our point of view a set is compact if and only if it is bounded and closed. Every sequence in a compact set has a convergent subsequence.

Functions of multiple variables

We would like to study functions of multiple variables, which means that the functions we will consider depend on more than one argument, e.g.:

    \[f(x,y)=x^2+2xy-3\]

is a function of two variables. Such functions obviously appear in the real world, e.g. temperature on a surface which depends on coordinates x,y. Or the height of a terrain.

Obviously, we can also easily check the domain of such a function. In the case of two arguments it will be a subset of a plane. E.g. the domain of f(x,y)=\frac{1}{x+y} is \{(x,y)\colon x+y\neq 0\}, so those are all the points of the plane except from the line y=-x.

Limits and continuity of functions

We say that a function f (assume that it depends on two arguments) has a limit in a point (x,y) equal to g, if for any sequence of points (x_n,y_n)\neq (x,y) in the domain, such that x_n\to x and y_n\to y, we get \lim_{n\to\infty}f(x_n,y_n)=g. But if for some of such sequences this limit does not exist or there are different limits for some different sequences, that the limit of the function in question does not exists. E.g. f(x,y)=xy has in (0,0) limit, because for any (x_n,y_n), such that x_n\to 0 and y_n\to 0, we have \lim_{n\to\infty}x_ny_n=0.

Similarly, as in the case of one-dimensional functions, a function is continuous in a point x_0 if its limit in this point equals its value. It is easy to see that it is equivalent to the property, that for every \varepsilon>0, there exists r>0 such that for every point x in the ball with centre in x_0 and radius r, f(x)\in(f(x_0)-\varepsilon, f(x_0)+\varepsilon).

A function is continuous if it is continuous at every point in the domain.