文档介绍：Chapter Seven
Continuity, Derivatives, and All That
Limits and Continuity
n n
Let x 0 Î R and r > 0. The set B(a;r) = {x Î R :|x - a|< r}is called the open
ball of radius r centered at x 0 . The closed ball of radius r centered at x0 is the set
B(a;r) = {x Î R n :|x - a|£ r}. Now suppose D Ì R n . A point a Î D is called an
interior point of D if there is an open ball B(a;r) Ì D . The collection of all interior
points of D is called the interior of D, and is usually denoted int D. A set U is said to be
open if U = int U.
Suppose f :D ® R p , where D Ì R n and suppose a Î Rn is such that every
open ball centered at a meets the domain D. If y Î R p is such that for every e > 0, there
is a d > 0 so that| f (x) - y|< e whenever 0 <| x - a|< d , then we say that y is the limit of
f at a. This is written
lim f (x) = y ,
x® a
and y is called the limit of f at a.
Notice that this agrees with our previous definitions in case n = 1 and p =1,2, or 3.
The usual properties of limits are relatively easy to establish:
lim( f (x) + g(x)) = lim f (x) + lim g(x) , and
x® a x® a x® a
lim af (x) = a lim f (x) .
x® a x® a
Now we are ready to say what we mean by a continuous function f :D ® R p ,
where D Ì R n . Again this definition will not contradict our previous lower dimensional
definitions. Specifically, we say that f is continuous at a Î D if lim f (x) = f (a) . If f is
x® a
continuous at each point of its domain D, we say simply that f is continuous.
Example
Every linear function is continuous. To see this, suppose f :Rn ® R p is linear
e
and a Î R n . Let e > 0. Now let M = max{| f (e )|,| f (e )|, ,| f (e )|}and let d = .
1 2 K n nM
Then for x such that 0 <|x - a|< d , we have
| f (x) - f (a)|=| f (x1e1 + x2e2 +K+xn en ) - f (a1e1 + a2e2 +K+a nen )|
=|( x1 - a1) f (e1 ) + (x2 - a2 ) f (e2 )+K+(xn - an ) f (en )|
£| x1 - a1|| f (e1 )|+|x2 - a2 || f (e2 )|+K+| xn - an || f (en )|
£ (| x1 - a1|+| x2 - a 2|+K+|xn -