Section 2.5: Continuity
Definition: A function f is continuous at x = a if
lim f (x) = f (a).
x→a
In order for f to be continuous at x = a, the following conditions must hold:
1. f (a) is defined
2. lim f (x) exists
x→a
3. lim f (x) = f (a).
x→a
If f is not continuous at x = a, we say f is discontinuous at x = a or has a discontinuity
at x = a. The three types of discontinuities are holes, jumps, and vertical asymptotes.
Example: Explain why each function is discontinuous at the given point.
(a) f (x) =
1
,x=2
(x − 2)2
(b) f (x) =
1 − x if x ≤ 2
x2 − 2x if x > 2.
1

 x2 − 2x − 8
if x 6= 4
(c) f (x) =
x
−
4

3
if x = 4.
Example: Find the points at which f is discontinuous

 2x + 1 if x ≤ −1
3x
if −1 < x < 1
f (x) =

2x − 1 if x ≥ 1.
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Example: Find the constant c that makes g continuous on (−∞, ∞)
g(x) =
x2 − c2 if x < 4
cx + 20 if x ≥ 4.
Example: Find the values of a and b that make

 2x
ax2 + b
f (x) =

4x
3
f continuous on R.
if x < 1
if 1 ≤ x ≤ 2
if x > 2.
Definition: A function f is continuous from the right at x = a if
lim f (x) = f (a).
x→a+
Similarly, f is continuous from the left at x = a if
lim f (x) = f (a).
x→a−
Example: The graph of a function f is given below. State the numbers at which f is
discontinuous. For each point of discontinuity, state whether f is continuous from the right,
left, or neither.
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Definition: A function f is said to have a removable discontinuity at x = a if lim f (x)
x→a
exists, but is not equal to f (a). The discontinuity can be removed by redefining f (a) so that
lim f (x) = f (a).
x→a
Example: Which of the following functions f has a removable discontinuity at x = a? If the
discontinuity is removable, find a function g that agrees with f for x 6= a and is continuous
on R.
(a) f (x) =
x2 + 2x − 8
,a=2
x−2
(b) f (x) =
x−7
,a=7
|x − 7|
Theorem: (Intermediate Value Theorem)
If f is continuous on the closed interval [a, b] and N is any number strictly between f (a) and
f (b), then there exists a number c in (a, b) such that f (c) = N .
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Example: Show that there is a root of the equation 4x3 − 6x2 + 3x − 2 on the interval (1, 2).
Example: If f (x) = x3 − x2 + x, show that there is a number c such that f (c) = 10.
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