Download 1.4 One-Sided Limits and Continuity

Sect.1.4 continued
One-Sided Limits and
Continuity
Continuity
A function is continuous at c if the
following three conditions are met
1. f(c) is defined
2. Limit of f(x) exists
lim f ( x)  lim f ( x)
x n
xn
3. Limit of f(x) is f(c)
Properties of Continuity
If b is a real number and f ( x) and g ( x) are both
continuous at x = c then, the following are also
continuous
b f
1.
Scalar:
2.
Sum/Difference:
3.
Product:
f g
4.
Quotient:
f
g
5.
Composite: If g(x) is continuous at c and f(x) at g(c),
f g
then f[g(x)] is also continuous at c
 x 2  2 x,
8) Show that f ( x)   x 3  6 x,

x  2
is
x  2
continuous at x = –2
Check the three conditions
a. f (2) exists
?
b. lim  f ( x) 
x 2
lim  f ( x)
x  2
Jump Discontinuity
3x  5, x  1
9) Determine if f (x)  
x 1
2,
continuous for x = 1
Check the three conditions
a.
b.

f (1)
exists
?
lim f ( x)  lim f ( x)
x 1
x 1
?
c. lim f ( x)  f (1)
x 1
is
ax  5,
10) For what value of ‘a’ is f ( x)   2
 x  x,
continuous?
x4
x4
x  1
2,

f ( x)  ax  b, - 1  x  3
11)
is continuous for
 2,
x3

all real numbers for what value of ‘a’ and ‘b’
when x  1
a. f (1) exists
when x  3
a. f (3) exists
12)
x3  1
f ( x)  
x  1
x 1
is it continuous at
x 1 ?
a. f (1) exists
x 1
?
c. f ( x)  lim f ( x)
x1
b.
?
lim f ( x)  lim f ( x)
x 1
x 1
HOMEWORK
• Page 80 # 35-53 odd, 63-66 all