1549901369-Elements_of_Real_Analysis__Denlinger_

4.2 Algebra of Limits of Functions 189

t t t

x 0 -e Xo x 0 +e

Figure 4.3

The language of neighborhoods and deleted neighborhoods provides a useful way

of verbalizing the limit statement. If x 0 is a cluster point of V(J),

lim f(x) = L ¢=? Ve> 0, :Jo> 0 3 x E N{;(xo) n 'D(J) =? f(x) E Ng(L)

X--+Xo

{:} V nbd. N of L , :J deleted nbd. M of xo 3 f(M) i;;; N

{:} V nbd. N of L , :J deleted nbd. M of x 0 3M i;;; f-^1 (N)

{:} the inverse image of every neighborhood of L contains a

deleted neighborhood of xo.

Definition 4.2.4 A function f is said to be constant on a set A if :Jc E JR 3
'v'x E A,f(x) = c.

Theorem 4.2.5 If f is constant, say f(x) = c, on some deleted neighborhood

of xo, then lim f(x) = c.

X--+Xo

Proof. Exercise 2. •

Definition 4.2.6 A function f is said to be bounded on a set Ai;;; V(J) if
:JB > 0 3 'v'x EA, lf(x)I::; B. [Equivalently, :Ja,b E JR 3 'v'x EA, a::; f(x)::;
b.]

Theorem 4.2.7 If lim f(x) =LE JR, then there is some neighborhood N of

x--+xo

xo such that f is bounded on N n 'D(J).

Proof. Suppose lim f(x) = L E R Letting c = 1 in Definition 4.1.1,

X--+XQ

:Jo > 0 3 'v'x E 'D(J),

0 < Ix -xol < o =? lf(x) -LI < 1

=* -1 < f(x) - L < 1

=? L - 1 < f(x) < L + l.

Therefore, f is bounded on N 0 (xo) n 'D(J). •