Advanced Calculus Topics - Computer Graphics and Geometric Modeling - page 214

1
2

p

Ó ˛

( )

O = r cos , sin q r q << r 2 and 0 << q .

2

See Figure 4.4.
It is easy to show that the relative concepts of open and closed do not depend on
R n . Specifically, if X is also a subset of R m , then A will be open/closed in X with respect
to R n if and only if it is open/closed with respect to R m . Although we shall not give
the formal definitions, we now also get the obvious relative concepts of limit point of
A in X , closure of A in X , and boundary of A in X .
Next, we turn to the topology of continuity. Invariably, whenever one defines some
structure in mathematics it is useful to define maps (“morphisms”) that preserve this
structure. Since the only topological structure that we have on R n at the moment is
that of open sets, it is natural to define these maps in terms of them. We begin with
a notion of limits.

Definition. We say that a sequence of points p 1 , p 2 ,... has a limit point p , and write

lim pp ,

i

i Æ•

if for every e>0 there is an N so that i > N implies that

pp

i -< .

When it comes to functions, it is assumed that the reader has seen the usual e-d
type definition for continuity, at least in the case of real-valued functions of one vari-
able. That definition extends to functions of more variables almost verbatim.
Let X Õ R m and Y Õ R n .

Definition. Let f : X Æ Y and let p Œ X . We say that the function f has a limit L at p

and write

() =

lim f qL ,

qp Æ

if L ΠY and for every e>0 there is a d>0 so that 0 <| q - p |<dand q ΠX implies
that

O

S 1

A

1/2 1 2

Figure 4.4. An open set in the circle.