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Section 2.5: Continuity
Definition We say f (x) is continuous at x = a if lim f (x) = f (a). Note that in order for this
x→a
definition to be met, the following conditions must hold:
(a) x = a is in the domain of f (x) (This ensures that f (a) is defined).
(b) lim f (x) must exist. Thus lim− f (x) = lim+ f (x).
x→a
x→a
x→a
(c) lim f (x) = f (a).
x→a
Some types of discontiniuties include ‘holes’, ‘jumps’ and ‘vertical asumptotes’.
The graphs below represents a discontinuity at x = 4 because f (4) is not defined.
The graphs below represents a discontinuity at x = 4 because lim f (x) does not exist
x→4
2
The graphs below represents a discontinuity at x = 4 because lim f (x) 6= f (4)
x→4
EXAMPLE 1: For the graph of f (x) given below, locate all discontiniuties. For each discontinuity,
find the limit from the left and the limit from the right.
7
f(x)
6
5
4
3
2
1
−7 −6 −5 −4 −3 −2 −1
1
−1
−2
−3
−4
−5
−6
2
3
4
5
6
7
3
EXAMPLE 2: Explain why the following functions are or are not continuous at the indicated value
of x. Support your answer.
−1
(i) f (x) =
, x = 1.
(1 − x)2

x2 − 2x − 8



if x 6= 4
x−4
at x = 4
(ii)f (x) =



3
if x = 4

2x + 1
if x ≤ −1



3x
if − 1 < x < 1
(iii) f (x) =
at x = ±1
2
x +2
if x > 1



4
if x = 1
4
x2 − c2 if x < 4
For what value(s) of c is g(x) continuous? In order to
cx + 20 if x > 4
receive full credit, your answer must be fully supported by the definition of continuity.
EXAMPLE 3: If g (x) =
 2
 x + c if x > 1
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if x = 1 ,For what value(s) of c is f (x) continuous, if any? In
EXAMPLE 4: If f (x) =

4cx + 3 if x < 1
order to receive full credit, your answer must be fully supported by the definition of continuity.
Definition We say f (x) is continuous from the right at x = a if lim+ f (x) = f (a). Similarly, we say
x→a
f (x) is continuous from the left at x = a if lim− f (x) = f (a).
x→a
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Intermediate Value Theorem If f (x) is continous on the interval [a, b] and N is any number
strictly between f (a) and f (b), then there is a number c, a < c < b, so that f (c) = N .
EXAMPLE 5: If g(x) = x5 − 2x3 + x2 + 2, show there a number c so that g(c) = −1.
EXAMPLE 6: Show there is a root to the equation x5 − 2x4 − x − 3 on the interval (2, 3).