ANALYSIS I
Limits of Functions and Continuity
 Handout #5 - 2/19/96 

Defn. A point x0 is called a limit point of a set A if each nbhd of x0 contains a member of A different from x0, i.e. for each e > 0,   (Ne(x0) \{x0}) Ç A ¹Æ.

Defn. A point x0 Î A is called an isolated point of A if x0 belongs to A but is not a limit point.

Theorem. A set F is closed if and only if it contains all its limit points.

Theorem. x0 is a limit point of a set A if and only if there exists a sequence {xn} Í A such that xn ® x0, but xn¹ x0, ("n Î IN).

Defn. Suppose that x0 is a limit point of the domain of a function f, then f is said to have a limit L as x® x0  if,

"e > 0, $d > 0   '  ( x Î dom(f)  &  0 < |x-x0| < d) Þ |f(x)-L| < e.
In this case, we use the notation,

lim
x® x0		 f(x) = L.

Theorem.  Suppose that f: A® B is a real-valued function of a real variable, i.e. A,B Í IR. If x0 is a limit point of the domain of f, then TFAE (The Following Are Equivalent):

a.) limx® x0 f(x) = L,
b.) For every sequence { xn} in the domain of f, if xn ®x0, then f(xn) ® L.
Defn. Suppose A,B Í IR and f: A® B. If x0 Î A, then f is said to be continuous at x0 if either
  1. x0 is an isolated point of A

    2. or
  2. limx® x0 f(x) = f(x0).
Defn.  Consider a set B Í IR. A set  O~Í B is called open relative to B (or briefly, relatively open) if  O~ =  OÇB for some open set  O Í IR.

Theorem.  Suppose that f:A® B, where A,B Í IR, then TFAE:

a.)   f is continuous at each point of its domain,
b.)   for each x0 Î A and for every e > 0, there is a d > 0

        such that whenever |x-x0| < d, then |f(x)-f(x0)| < e,
c.)   if xn® x0, then f(xn) ® f(x0),
d.)   if f-1[ O] is open for each open subset O of B.
Corollary. The finite sum, product, or the quotient of continuous functions is each continuous on their respective domains.

Corollary. All polynomials are continuous. Rational functions are continuous on their domains.

Theorem. The composition of continuous functions is continuous.

Examples. Each of the following are examples of continuous functions:

  1. f(x): = |x|
  2. g(x): = Öx
  3. F(x): = Ö{[(x2-2x+5)/(x3-1)]}

    4.  
Homework #7 (Due Friday, Feb. 23)
  1. Suppose that x0 is a limit point of the domain of both f and g have limits at x0, then prove that

    2. lim
    x® x0    		 (f+g)(x) = 
    lim
    x® x0    		 f(x) + 
    lim
    x® x0    		 g(x).
  2.  Suppose that f is defined by
    3. f(x) : =  ì
    í
    î
    3x+2, 
    if   -1 £ x
    -2x+1, 
    if   x < -1.
    Determine at each point whether or not f is continuous. Justify your answer.
  3.   Determine the domain of F(x): = Ö{[(x2-2x+5)/(x3-1)]} and carefully show that F is continuous on its domain.

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