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Continuity and OneSided Limits (1.4)
September 9th, 2016
I. Continuity at a Point on an Open
Interval
Def: A function is continuous at a point c if
1. f(c) is defined
lim f (x)
2.
exists, and
xc
lim f (x)  f (c) .
3.
xc
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.
(,)
If a function f is continuous on the open interval
(a, b) except at point c, it is said to have a
discontinuity at c. This discontinuity is removable
if f can be made continuous by appropriately
defining (or redefining) f(c). Otherwise, the
discontinuity is nonremovable.
f(c) is not
defined
Removable
discontinuity
at c.
Nonremovable
discontinuity
at c. (This is
known as a
jump
discontinuity)
Another type
of
nonremovable
discontinuity is
an asymptote.
lim f (x)
xc
does not exist
lim f (x)  f (c)
xc
Removable
discontinuity at
c.
Ex. 1: Find all values of x for which the function is
discontinuous. Describe each of the
discontinuities as removable/nonremovable in
addition to identifying whether the discontinuity is
a jump discontinuity, an asymptote, or a hole.
(a)
(b)
x 2  6x  5
f (x) 
x5
x 2  2x 1
g(x) 
x 1
(c)
 x 2  3, x  1

h(x)  2,1  x  1
2x  3, x  1

(d)
sec x
k(x) 
x
II. One-Sided Limits and Continuity on
a Closed Interval
One-Sided Limits are denoted by the following.
means the limit as x
lim f (x)
approaches
xc
lim f (x)
xc
c from the right, and
means the limit as x
approaches
c from the left.
Thm. 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 if and only if
lim f (x)  L
xc
and
lim f (x)  L
xc
Ex. 2: Find each limit (if it exists). If it does not
a) lim f (x)
exist, explain why.
x5
b) lim f (x)
x5
c) lim f (x)
x4
d) lim f (x)
x4
e) lim f (x)
x4
f ) lim f (x)
x2
g)lim f (x)
x0
h) lim f (x)
x1
i) lim f (x)
x1
j)lim f (x)
x1
k)lim f (x)
x3
l) lim f (x)
x3
m) lim f (x)
x3
Def: A function f is continuous on a closed interval
[a, b] if it is continuous on the open interval (a, b)
and
and lim f (x)  f (b)
lim f (x)  f (a)
xb
xa
.
We say that f is continuous from the right of a and
continuous from the left of b.


Ex. 3: Give each of the intervals of x for which
the following function is continuous.
 1
 x 2  4 , x  1
g(x)  
 x , x  1
 3
III. Properties of Continuity
Thm. 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 and difference: f  g
3. Product: fg
f
, g(c)  0
4. Quotient:
g
Functions that are Continuous at Every Point in
their Domain:
n
n1
p(x)

a
x

a
x
 ... a1x  a0
n
n1
1. Polynomial functions
2. Rational functions
3. Radical functions
p(x)
r(x) 
,q(x)  0
q(x)
f (x)  n x
4. Trigonometric functions
sin x,cos x, tan x,csc x,sec x,cot x
Thm. 1.12: Continuity of a Composite Function: If
g is continuous at c and f is continuous at g(c),
( f og)(x)
then the composite function given
by  f (g(x))
is continuous at c.
IV. The Intermediate Value Theorem
***Thm. 1.13: The Intermediate Value Theorem: If
f is continuous on the closed interval [a, b] and k
is any number between f(a) and f(b), then there is
at least one number c in [a, b] such that f(c)=k.
Ex. 4: Prove that the function f (x)  3sin x cos x
has a value of x such thatf (x)  1 4
in the
  
interval
.
,
 6 2 
Ex. 5: Water flows into a cylindrical tank
continuously for 12 hours, as given by the table
below. What is the least number of times within
those 12 hours that the rate of the water flow is
10 gallons/minute? Justify your reasoning.
time, in
hours
0
2
5
6
9
12
water
flow, in
gallons/
minute
14
9
7
12
16
9