Students
2024
CALCULUS 110
( 𝟗 𝑬)
(2.5) Continuity
24 August 2023
Homework
Examples
3,5,19,22,
23,27,28,34,41,45,47 2(a-c), 4-9 9(b) Change the
function to ln (x^2-25)
1 Dr. Rola Asaad Hijazi August
students
24, 2023
2.5 Continuity
A continuous process is one that takes place gradually, without interruption or abrupt
change.
Continuous at a number 𝑎
A function 𝑓 is continuous at a number 𝑎 if:
𝑙𝑖𝑚 𝑓 𝑥 = 𝑓 𝑎
𝑥→𝑎
So 𝑓 𝑥 is continuous at 𝑎 if:
1. 𝑓 𝑎 is defined 𝑎 ∈ 𝐷𝑓
2. 𝑙𝑖𝑚 𝑓 𝑥 exists
𝑥→𝑎
3. 𝑙𝑖𝑚 𝑓 𝑥 = 𝑓 𝑎
𝑥→𝑎
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Example 2 Where are each of the following functions discontinuous?
𝑥2 − 𝑥 − 2
a 𝑓 𝑥 =
𝑥−2
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Example 2 Where are each of the following functions discontinuous?
1
b 𝑓 𝑥 = ቐ 𝑥2 , 𝑥≠0
1, 𝑥=0
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Example 2 Where are each of the following functions discontinuous?
𝑥2 − 𝑥 − 2
c 𝑓 𝑥 =ቐ 𝑥−2 , 𝑥≠2
1 , 𝑥=2
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Definition (2) Definition (3)
continuous from the right
A function 𝑓 is continuous on an interval
A function 𝑓 is continuous from if it is continuous at every number in the
the right at a number 𝑎 if :
interval. (If 𝑓 is defined only on one side
𝑙𝑖𝑚 𝑓 𝑥 = 𝑓 𝑎
𝑥→𝑎+
of an endpoint of the interval, we
continuous from the left understand continuous at the endpoint
to mean continuous from the right or
And 𝑓 is continuous from the
left at 𝑎 if : continuous from the left.)
𝑙𝑖𝑚 𝑓 𝑥 = 𝑓 𝑎
𝑥→𝑎−
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Examples
(1) The function 𝑓 𝑥 = 3 − 𝑥 is continuous from the right at 𝑥 = 3.
(a) True
(b) False
(2) The function 𝑓 𝑥 = 𝑥 − 3 is
(a) continuous at 𝑥 = 3.
(b) continuous from the right at 𝑥 = 3.
(c) continuous from the left at 𝑥 = 3.
(d) continuous at 𝑥 = 0.
8 students August 24, 2023
Examples
𝑥 + 1 𝑖𝑓 𝑥 ≥ 2
(3) The function 𝑓 𝑥 = ቊ 2 is
𝑥 𝑖𝑓 𝑥 < 2
(a) continuous from the right at 𝑥 = 2.
(b) continuous from the left at 𝑥 = 2.
(c) Continuous at 𝑥 =ℝ
(d) continuous at 𝑥 = 2.
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Show that the function𝑓 𝑥 = 1 − 1 − 𝑥 2 is continuous on
Example 4
the interval [−1,1].
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Theorem 4 Theorem 5
If 𝑓 and 𝑔 are continuous at 𝑎 and 𝑐 is Any polynomial is continuous everywhere,
a constant, then the following that is , it is continuous on ℝ = −∞, ∞ .
functions are also continuous at 𝑎:
1. 𝑓 + 𝑔 Any rational function is continuous wherever
2. 𝑓 ⋅ 𝑔 it is defined, that is, it is continuous on its
3. 𝑐𝑓 domain.
4. 𝑓 − 𝑔
𝑓
5. 𝑔 (if 𝑔 𝑎 ≠ 0) Theorem 7
The following types of functions are
continuous at every number in their
Theorem 6 domains:
Polynomials
𝑙𝑖𝑚 𝑐𝑜𝑠 𝜃 = 1
𝜃→0 Trigonometric functions
Exponential functions
𝑙𝑖𝑚 𝑠𝑖𝑛 𝜃 = 0 Rational functions
𝜃→0
Root functions
Inverse trigonometric functions
Logarithmic functions
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𝑥 3 + 2𝑥 2 − 1
Example 5 Where is the following function 𝑓(𝑥) = continuous.
5 − 3𝑥
𝑥 3 + 2𝑥 2 − 1
Find 𝑙𝑖𝑚
𝑥→−2 5 − 3𝑥
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−1
Example 6 Where is the function 𝑓 𝑥 = 𝑙𝑛 𝑥 + 𝑡𝑎𝑛 𝑥 continuous?
𝑥2 − 1
solution
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𝑠𝑖𝑛 𝑥
Example 7 Where is the following function 𝑓 𝑥 = continuous.
2 + 𝑐𝑜𝑠 𝑥
𝑠𝑖𝑛 𝑥
𝑙𝑖𝑚
Evaluate 𝑥→𝜋 2 + 𝑐𝑜𝑠 𝑥
solution
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Theorem 8
If 𝑓 is continuous at 𝑏 and 𝑙𝑖𝑚 𝑔 𝑥 = 𝑏, then 𝑙𝑖𝑚𝑓 𝑔 𝑥 =𝑓 𝑏 .
𝑥→𝑎 𝑥→𝑎
i.e. 𝑙𝑖𝑚 𝑓 𝑔 𝑥 = 𝑓 𝑙𝑖𝑚 𝑔 𝑥
𝑥→𝑎 𝑥→𝑎
Theorem 9
If 𝑔 is continuous at 𝑎 and 𝑓 is continuous at 𝑔 𝑎 , then the composite
function 𝑓 ∘ 𝑔 given by (𝑓 ∘ 𝑔)𝑥 = 𝑓 𝑔 𝑥 is continuous at 𝑎.
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Example8 Theorem 8
1− 𝑥 𝑙𝑖𝑚 𝑓 𝑔 𝑥 = 𝑓 𝑙𝑖𝑚 𝑔 𝑥
Evaluate 𝑙𝑖𝑚 𝑎𝑟𝑐𝑠𝑖𝑛 𝑥→𝑎 𝑥→𝑎
𝑥→1 1−𝑥
solution
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Example 9 Where is the following function continuous?
a ℎ 𝑥 = 𝑠𝑖𝑛 𝑥 2
solution
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Example 9 Where is the following function continuous?
b 𝐹 𝑥 = 𝑙𝑛 𝑥 2 − 25
solution
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Exercise 29
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at
every number in its domain. State the domain.
cos(𝑡 2 )
ℎ 𝑡 =
1 − 𝑒𝑡
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homework
3,5,19,22, 23,27,28,34,41,45,47
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