Download 2.3 Continuity Continuity of a Function at a Point A function f is

2.3
Continuity
Continuity of a Function at a Point
A function f is continuous at a point x = c if the following are satisfied:
1.
f(c) is defined
2.
lim f(x) exists
3.
lim f(x) = f(c)
xc
xc
A function that is not continuous at c is said to have a discontinuity at that
point.
Example: Determine whether or not the function f(x) =
x2  2x  1
is
x3
continuous at x = 3.
Example: Determine whether or not the function f(x) is continuous at x =
x2  4 x  3
3, f(x) =
if x ≠ 3 and f(x) = 2 if x = 3.
x3
Continuity Theorem
If f is a polynomial or a rational function, a power function, a trigonometric
function or an inverse trigonometric function, then f is continuous at any
number x = c for which f(c) is defined.
Properties of Continuous Functions
If f and g are functions that are continuous at x = c, then the following
functions are also continuous at x = c.
Scalar Multiple
sf
Sum
Difference
f+g
f–g
Product
fg
Quotient
f
g
Composition
f°g
Composition Limit Rule
If lim g(x) = L and f is a function continuous at L, then
xc
lim f(g(x)) = f(L)
xc
One-Sided Continuity
The function f is continuous from the right at a iff
lim f(x) = f(a)
x  a
The function f is continuous from the left at b iff
lim f(x) = f(b)
x b 
Continuity on an Interval
A function f is continuous on the open interval (a,b) if it is continuous at
each number in this interval. A function f is continuous on the closed
interval [a,b] if it is continuous at each number between a and b and it is
continuous from the right at a and continuous from the left at b.
When determining intervals of continuity of a function, look for suspicious
points. A suspicious number is a number x = c where either
1.
The defining rule for f changes.
2.
Substitution of x = c causes division by 0 in the function.
Example: Determine if the function f(x) is continuous on the interval (0,5),
if
 x2
if 0  x  2
f(x) = 
 3x  1 if 2  x  5
Example: Determine if the function f(x) is continuous on the interval (0,)
if
f(x) = x sin x
The Intermediate Value Theorem
If f is a continuous function on the closed interval [a,b] and L is some
number strictly between f(a) and f(b), then there exists at least one
number c on the open interval (a,b) such that f(c) = L.
Root Location Theorem
If f is continuous on the closed interval [a,b] and f(a) and f(b) have
opposite signs, then f(c) = 0 for at least one number c on the open interval
(a,b).
Example: Show that the equation has at least one solution on the interval
(1,2):
1
= x2 – x - 1
x 1
Example: The population (in thousands) of a colony of bacteria t minutes
after the introduction of a toxin is given by the function
 t2  1
if 0  t  5
P(t) = 
  8t  66 if t  5
a)
When does the colony die out?
b)
Show that at some time between t = 2 and t = 7, the population is
9,000.