MATH 131-505 Spring 2015
c
Wen
Liu
2.4
2.4 Continuity
A function f is continuous at a number a if
lim f (x) = f (a)
x→a
Notice that this definition implicitly requires three things if f is continuous at a:
1. f (a) is defined (that is, a is in the domain of f )
2. lim f (x) exists
x→a
3. lim f (x) = f (a)
x→a
If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps
at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not continuous at a.
Examples:
1. (p. 113) The graph of a function f is shown below. At which numbers is f discontinuous? Why?
2. (p. 114) Where are each of the following functions discontinuous?

 1 , if x 6= 0
x2 − x − 2
(b) f (x) = x2
(a) f (x) =
1,
x−2
if n = 0
 2
 x − x − 2 , if x 6= 2
x−2
(c) f (x) =

1,
if x = 2
(d) f (x) = JxK, the greatest integer function.
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MATH 131-505 Spring 2015
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c
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Liu
• The kind of discontinuity illustrated in parts (a) and (c) is called removable because we could
remove the discontinuity by redefining f at just the single number 2.
• The kind of discontinuity illustrated in part (b) is called an infinite discontinuity.
• The kind of discontinuities illustrated in part (d) are called jump discontinuities because the
function “jumps” from one value to another.
Example 3: (p. 121) Sketch the graph of a function f that is continuous except for the stated
discontinuity: removable discontinuity at 3 and jump discontinuity at 5.
A function f is continuous from the right at a number a if
lim f (x) = f (a)
x→a+
and f is continuous from the left at a if
lim f (x) = f (a)
x→a−
A function f is continuous on an interval if it is continuous at every number in the interval. (If f
is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint
to mean continuous from the right or continuous from the left.)
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MATH 131-505 Spring 2015
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c
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Liu
Examples:
4. Find each x-value at which f is discontinuous and for each x-value, determine whether f is
continuous from the right, or from the left, or neither. Sketch the graph of f .


x + 3,
if x ≤ 1


1
f (x) =
,
if 1 < x < 3

x

√

x − 3, if x ≥ 3
5. (p. 121) Sketch the graph of a function f that is continuous except for the stated discontinuity:
discontinuous, but continuous from the right, at 2.
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MATH 131-505 Spring 2015
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Liu
2.4
Theorem: If f and g are continuous at a and c is a constant, then the following functions are also
continuous at a:
f
f ± g, cf, f g,
if g(a) 6= 0
g
Examples:
√
40 + x
6. Use continuity to evaluate lim √
.
x→9
40 + x
7. Suppose f and g are continuous functions such that g(7) = 2 and lim (3f (x) + f (x)g(x)) = 15.
x→7
Find f (7).
Theorem: The following types of functions are continuous at every number in their domains:
• polynomials
• rational functions
• root functions
• trigonometric functions
• exponential functions
• logarithmic functions
Examples:
8. Locate the discontinuities of the function y =
5
.
2 + e2/x
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MATH 131-505 Spring 2015
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Liu
9. (p. 122) Find the values of a and b that make f continuous on (−∞, ∞)?
 2

x − 4,
if x < 2

 x−2
f (x) = ax2 − bx + 3, if 2 ≤ x < 3



2x − a + b,
if x ≥ 3
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