Section 3.2: 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.
2
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.
4