CONTINUITY OF
FUNCTION
If
LIMITS OF A FUNCTION
CONTINUITY OF FUNCTION
Continuity
Informally, a continuous function is
one whose graph If can be drawn
without the “pen” leaving the paper.
One with no holes or gaps.
LIMITS OF A FUNCTION
TYPES OF DISCONTINUITY
1. Essential Discontinuity
A function f(x) is said to have essential discontinuity at x = c if
𝐥𝐢𝐦 𝐟(𝐱) does not exist.
𝐱→𝐜
𝑥+1 𝑖𝑓 𝑥 < 4
𝒉 𝒙 =ቊ 2
If 𝑥−4 𝑖𝑓 𝑥 ≥ 4
LIMITS OF A FUNCTION
TYPES OF DISCONTINUITY
2. Removable Discontinuity
A function f(x) is said to have removable discontinuity at x = c if:
• 𝐥𝐢𝐦 𝐟(𝐱) exist; and
𝐱→𝐜
• either f(c) does not exist or
f(c) ≠ 𝐥𝐢𝐦 𝐟(𝐱)
𝐱→𝐜
If
3𝑥 2 − 4𝑥 + 1
𝒈 𝒙 =൞ 𝑖𝑓 𝑥 ≠ 1
𝑥−1
1 𝑖𝑓 𝑥 = 1
LIMITS OF A FUNCTION
TYPES OF DISCONTINUITY
ILLUSTRATION
Evaluate if the graph is continuous at:
x=1 discontinuous
x If= 2 discontinuous
x=3 discontinuous
x=4 continuous
x=5 discontinuous
x=6 continuous
LIMITS OF A FUNCTION
CONTINUITY TEST
A function f is continuous at the number a if and only if the
following three conditions are ALL satisfied:
i. f(a) exists
ii. 𝑙𝑖𝑚 𝑓(𝑥) exists If
𝑥→𝑎
iii. 𝑙𝑖𝑚 𝑓(𝑥) = f(a)
𝑥→𝑎
If one of these conditions fails to hold at a, then f is said to be
discontinuous at a.
LIMITS OF A FUNCTION
EXAMPLES:
Determine if the following functions are continuous at:
i. p(x) = 3𝑥 2 − 𝑥 + 5; 𝑎𝑡 𝑥 = 1
p(1) = 3(1)2 −(1) + 5
p(1) = 3 − 1 + 5 If
p(1) = 𝟕
LIMITS OF A FUNCTION
EXAMPLES:
Determine if the following functions are continuous at:
ii. 𝑙𝑖𝑚 (3𝑥 2 − 𝑥 + 5)
𝑥→1
= (3 1 2 − (1) + 5)
= (3(1) − 1 + 5) If
= 𝟕
iii. 𝑝 1 = 𝑙𝑖𝑚 𝑝(𝑥)
𝑥→1
= 𝑝 𝑥 is continuous at x = 1
LIMITS OF A FUNCTION
EXAMPLES:
Determine if the following functions are continuous at:
2
i. f(x) = ; 𝑎𝑡 𝑥 = 0 𝑎𝑛𝑑 𝑥 = −1
𝑥+1
2
f(0) =
(0)+1 If
2
f(0) =
1
f(0) = 𝟐
LIMITS OF A FUNCTION
EXAMPLES:
Determine if the following functions are continuous at:
2
ii. 𝑙𝑖𝑚 iii. 𝑓 0 = 𝑙𝑖𝑚 𝑓(𝑥)
𝑥→0 𝑥+1 𝑥→0
2 = f 𝑥 is continuous at x = 0
= If
(0)+1
2
=
1
=𝟐
LIMITS OF A FUNCTION
EXAMPLES:
Determine if the following functions are continuous at:
2
i. f(x) = ; 𝑎𝑡 𝑥 = 0 𝑎𝑛𝑑 𝑥 = −1
𝑥+1
2
f(-1) =
(−1)+1 If
2
f(0) =
0
f(0) = 𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅
f(x) is discontinuous at x = -1
LIMITS OF A FUNCTION
EXAMPLES:
Determine if the following functions are continuous at:
2𝑥
i. g(x) = 2 ; 𝑎𝑡 𝑥 = −1 𝑎𝑛𝑑 𝑥 = 3
𝑥 −4𝑥+3
2(−1)
g(-1) =
(−1)2 −4(−1)+3 If
−2
g(-1) =
1+4+3
−2
g(-1) =
8
𝟏
g(-1) = −
𝟒
LIMITS OF A FUNCTION
EXAMPLES:
Determine if the following functions are continuous at:
2𝑥
ii 𝑙𝑖𝑚 2 iii. 𝑔 −1 = 𝑙𝑖𝑚𝑔(𝑥)
𝑥→−1 𝑥 −4𝑥+3 𝑥→−1
2(−1)
= = g 𝑥 is continuous at x = -1
(−1)2 −4(−1)+3 If
−2
=
1+4+3
−2
=
8
𝟏
=−
𝟒
LIMITS OF A FUNCTION
THEOREM ON
CONTINUITY OF
POLYNOMIAL AND
If
RATIONAL FUNCTIONS
LIMITS OF A FUNCTION
THEOREMS ON CONTINUITY OF FUNCTIONS
a. A polynomial function is continuous for all x.
𝑝(𝑥)
b. A rational function If where p(x) and q(x)
𝑞(𝑥)
are polynomials is continuous for all x for
which q(x) ≠ 0.
LIMITS OF A FUNCTION
THEOREMS ON CONTINUITY OF FUNCTIONS
Determine the point(s) of discontinuity given the following
functions:
1
1. f(x) = f(x) is discontinuous at x = 0
𝑥
𝑥 2 −1 If
2. g(x) = g(x) is discontinuous at x = -1
𝑥+1
x + 1 = 0,
x = -1
LIMITS OF A FUNCTION
THEOREMS ON CONTINUITY OF FUNCTIONS
Determine the point(s) of discontinuity given the following
functions:
𝑥
3. h(x) = f(x) is discontinuous at x = 4 and x = 3
𝑥 2 −7𝑥+12
𝑥 2 − 7𝑥 + 12 = 0 If
(x - 4)(x - 3) = 0
x–4=0 x–3=0
x=4 x=3
LIMITS OF A FUNCTION
THEOREMS ON CONTINUITY OF FUNCTIONS
Determine the point(s) of discontinuity given the following functions:
𝑥 + 1; 𝑥 < 1
4. m(x) = ቊ m(x) is discontinuous at x = 1
2 − 𝑥; 𝑥 ≥ 1
m(1) = 2 – x
m(1) = 2 -1 If
m(1) = 1
𝑙𝑖𝑚− 𝑥 + 1 = 1 + 1 = 2
𝑥→1
𝑙𝑖𝑚+ 2 − 𝑥 = 2 − 1 = 1
𝑥→1
LIMITS OF A FUNCTION
CONTINUITY OF A
FUNCTION ON AN
If
INTERVAL
LIMITS OF A FUNCTION
CONTINUITY OF A FUNCTION ON AN INTERVAL
𝐄𝐗𝐀𝐌𝐏𝐋𝐄 𝐈𝐍𝐓𝐄𝐑𝐕𝐀𝐋 𝐒𝐄𝐓 𝐍𝐎𝐓𝐀𝐓𝐈𝐎𝐍 𝐆𝐑𝐀𝐏𝐇
(1, 2) (𝑎, 𝑏) {𝑥|𝑎 < 𝑥 < 𝑏} 𝑂𝑝𝑒𝑛 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
𝑎 𝑏
[1, 2] [𝑎, 𝑏] {𝑥|𝑎 ≤ 𝑥 ≤ 𝑏} 𝐶𝑙𝑜𝑠𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
𝑎 𝑏
[1, 2) [𝑎, 𝑏) {𝑥|𝑎 ≤ 𝑥 < 𝑏} 𝑎 𝑏
𝐶𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
(1, 2) (𝑎, 𝑏] {𝑥|𝑎 < 𝑥 ≤ 𝑏} 𝑎 𝑏
(1, ∞) (𝑎, ∞) {𝑥|𝑎 < 𝑥} 𝑎 𝑂𝑝𝑒𝑛 𝑙𝑒𝑓𝑡 − 𝑏𝑜𝑢𝑛𝑑𝑒𝑑
If
[1, ∞) [𝑎, ∞) {𝑥|𝑎 ≤ 𝑥} 𝑎
𝐶𝑙𝑜𝑠𝑒𝑑 𝑙𝑒𝑓𝑡 − 𝑏𝑜𝑢𝑛𝑑𝑒𝑑
(−∞, 2) (−∞, 𝑏) {𝑥|𝑥 < 𝑏} 𝑏
𝑂𝑝𝑒𝑛 𝑟𝑖𝑔ℎ𝑡 − 𝑏𝑜𝑢𝑛𝑑𝑒𝑑
(−∞, 2] (−∞, 𝑏] {𝑥|𝑥 ≤ 𝑏} 𝑏
𝐶𝑙𝑜𝑠𝑒𝑑 𝑟𝑖𝑔ℎ𝑡 − 𝑏𝑜𝑢𝑛𝑑𝑒𝑑
(−∞, ∞) 𝑅(𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠) 𝑈𝑛𝑏𝑜𝑢𝑛𝑑𝑒𝑑
LIMITS OF A FUNCTION
CONTINUITY OF A FUNCTION ON AN INTERVAL
If
LIMITS OF A FUNCTION
