Download 1.4 Continuity and One-Sided Limits Objective: Determine continuity

Miss Battaglia
AB/BC Calculus

What does it mean to be continuous?
Below are three values of x at which the graph of f is NOT continuous
At all other points in the interval (a,b), the graph of f is uninterrupted
and continuous
lim f (x)
x®c
f(c) is not
defined
does not exist
lim f (x) ¹ f (c)
x®c
Continuity at a Point: A function f is continuous at c if the
following three conditions are met.
1. f(c) is defined
2.
lim f (x) exists
3.
lim f (x) = f (c)
x®c
x®c
Continuity on an Open Interval: A function is continuous on an
open interval (a,b) if it is continuous at each point in the interval.
A function that is continuous on the entire real line (-∞,∞) is
everywhere continuous.

Removable (f can be made continuous by
appropriately defining f(c)) & nonremovable.
Nonremovable Discontinuity
Removable Discontinuity

Discuss the continuity of each function
1
f (x) =
x
x -1
g(x) =
x -1
2
ì x +1, x £ 0
ï
h(x) = í
ïî x 2 +1, x > 0
k(x) = sin x
• Limit from the right
lim+ f (x) = L
x®c
• Limit from the left
lim- f (x) = L
x®c
• One-sided limits are useful in taking
limits of functions involving radicals (Ex:
if n is an even integer)
lim+ n x = 0
x®0

Find the limit of f (x) = 4 - x 2 as x approaches
-2 from the right.

One sided limits can be used to investigate the behavior of step
functions. A common type is the greatest integer function 𝑥
defined by
𝒙 = greatest integer n such that n < x
◦ Ex: 2.5 = 2 and −2.5 = -3

Find the limit of the greatest integer function
𝑓 𝑥 = 𝑥 as x approaches 0 from the left and from
the right.
𝑓 𝑥 = 𝑥
Theorem 1.10: The Existence of a Limit
Let f be a function and let c and L be real numbers.
The limit of f(x) as x approaches c is L iff
lim- f (x) = L and
x®c
lim+ f (x) = L
x®c
Definition of Continuity on a Closed Interval
A function f is continuous on the closed interval
[a,b] if it is continuous on the open interval (a,b)
and
lim+ f (x) = f (a) and lim- f (x) = f (b)
x®a
x®b
The function f is continuous from the right at a and
continuous from the left at b.

Discuss the continuity of f (x) = 1- x 2
Theorem 1.11 PROPERTIES OF CONTINUITY
If b is a real number and f and g are continuous at x=c,
then the following functions are also continuous at c.
1. Scalar multiple: bf
2. Sum or difference: f + g
3. Product: fg
4. Quotient:
𝑓
,
𝑔
if g(c)≠0
By Thm 1.11, it follows that each of the following functions
is continuous at every point in its domain.
f (x) = x + sin x
f (x) = 3tan x
x +1
f (x) =
cos x
2
THEOREM 1.12 CONTINUITY OF A COMPOSITE FUNCTION
If g is continuous at c and f is continuous at g(c), then the
composite function given by 𝑓 ° 𝑔 𝑥 = 𝑓(𝑔 𝑥 ) is continuous at c.
Theorem 1.13 INTERMEDIATE VALUE THEOREM
If f is continuous on the closed interval [a,b], 𝑓(𝑎) ≠ 𝑓(𝑏)
and k is any number between f(a) and f(b), then there is at
least one number in c in [a,b] such that
f(c) = k


Consider a person’s height.
Suppose a girl is 5ft tall on her
thirteenth bday and 5ft 7in tall on
her fourteenth bday. For any
height, h, between 5ft and 5ft 7in,
there must have been a time, t,
when her height was exactly h.
The IVT guarantees the existence
of at least one number c in the
closed interval [a,b]

Use the IVT to show that the polynomial function
f(x)=x3 + 2x – 1 has a zero in the interval [0,1]
 AB:
Page 78 #27-30, 35-51 odd,
69-75 odd, 78, 79, 83, 91, 99-102
 BC:
Page 78 #3-25 every other odd,
31, 33, 34, 35-51 every other odd,
61, 63, 69, 78, 91,99-103