10/11/2024
Thomas’ Calculus: Early Transcendentals
Fifteenth Edition
Chapter 2
Limits and Continuity
Slide - 1
Section 2.6
Continuity
Examples Exercises
1,5, 6,7,8 (b,c) 1, 2, 4, 15
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Continuity at a Point 𝒂
Let 𝑐 be a real number that is either an interior point or an end point of
an interval in the domain of 𝑓.
The function 𝑓 is continuous at 𝑐 if
𝑙𝑖𝑚𝑓 𝑥 = 𝑓 𝑐
𝑥→𝑐
The function 𝑓 is right-continuous at 𝑐 (or continuous from the right) if
𝑙𝑖𝑚 𝑓 𝑥 = 𝑓 𝑐
𝑥→𝑐 +
The function 𝑓 is left-continuous at 𝑐 (or continuous from the left) if
𝑙𝑖𝑚 𝑓 𝑥 = 𝑓 𝑐
𝑥→𝑐 −
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Continuity test
1. 𝑓 𝑐 exists 𝑐 ∈ 𝐷𝑓
2. 𝑙𝑖𝑚𝑓 𝑥 exists (𝑓 has a limit as 𝑥 → 𝑐)
𝑥→𝑐
3. 𝑙𝑖𝑚𝑓 𝑥 = 𝑓 𝑐 ( (the limit equals the function value).
𝑥→𝑐
Continuity on the Interval
A function 𝑓 is continuous on an interval if it is continuous at every number
in the interval. (If 𝑓 is defined only on one side of an endpoint of the
interval , 𝑎, ∞ 𝑜𝑟 −∞, 𝑏 we study continuity from the right as 𝑥 → 𝑎+ or
from the left as 𝑥 → 𝑏 − ).
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𝑐 𝑐
𝒇(𝒄) is not defined 𝒍𝒊𝒎𝒇 𝒙 ≠ 𝒇 𝒄 Limit D.N.E.
𝒙→𝒄
5 students October 11, 2024
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Continuity on one sided Interval
𝑎, ∞
−∞, 𝑏
𝒍𝒊𝒎− 𝒇 𝒙 = 𝒇 𝒃 𝒍𝒊𝒎+ 𝒇 𝒙 = 𝒇 𝒂
𝒙→𝒃 𝒙→𝒂
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Examples on continuity from one side
(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.
7 students October 11, 2024
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Example 1
At which numbers does the function 𝑓 in the following Figure appear to be not
continuous? Explain why. What occurs at other numbers in the domain?
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Examples
𝑥 2 𝑖𝑓 𝑥 ≥ 0
1 The function 𝑓 𝑥 = ቊ
0 𝑖𝑓 𝑥 < 0
is
(a) continuous at 𝑥 = 0
(b) continuous on ℝ.
(c) continuous only on (−∞, 0) ∪ (0, ∞)
(d) continuous only on (0, ∞).
𝑥 − 1 𝑖𝑓 𝑥 < −1
2 The function 𝑓 𝑥 = ቐ2 𝑖𝑓 − 1 ≤ 𝑥 ≤ 1
𝑥+1 𝑖𝑓 𝑥 > 1
is discontinuous at
(a) −1,1 (b) 1
(c) −1 (d) −1,1
9 Dr. Rola Asaad Hijazi 08/04/1446
Example
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.
students
Exercise 31
At what points is the following function continuous?
1−𝑥, 𝑥<0
𝑓 𝑥 = ቐ𝑒 𝑥 , 0≤𝑥≤1
𝑥 2 + 2, 𝑥>1
11 students October 11, 2024
Continuity on the closed Interval
Continuity at points 𝒂, 𝒃 and 𝒄
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Example 2 Show that the function𝑓 𝑥 = 4 − 𝑥 2 is continuous on the
interval [−2,2].
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Exercises 1, 2 and 4
Say whether the function graphed is continuous on [−1, 3] . If not,
where does it fail to be continuous and why?
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Continuous Functions
We now describe the continuity behavior of a function throughout its
entire domain, not only at a single point. We define a continuous
function to be one that is continuous at every point in its domain. This
is a property of the function. A function always has a specified domain,
so if we change the domain, then we change the function, and this
may change its continuity property as well. If a function is
discontinuous at one or more points of its domain, we say it is a
discontinuous function.
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Theorem 8 Properties of Continuous Functions
If the functions 𝑓 and 𝑔 are continuous at 𝑥 = 𝑐, then the
following algebraic combinations are continuous at 𝑥 = 𝑐.
1. 𝑓 + 𝑔
2. 𝑓 ⋅ 𝑔
3. 𝑘𝑓
4. 𝑓 − 𝑔
𝑓
5. (if 𝑔 𝑐 ≠ 0)
𝑔
6. 𝑓 𝑛 𝑛 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟.
7. 𝑛 𝑓 provided it is defined on an interval containing c, where 𝑛 is a positive integer
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The following types of functions are continuous at every
number in their domains:
Polynomials
Trigonometric functions
Exponential functions
Rational functions
Root functions
Inverse trigonometric functions
Logarithmic functions
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Example 5
1
(a) Where is the function 𝑓 𝑥 = 𝑥 is continuous.
(b) Where is the function 𝑓 𝑥 = 𝑥 is continuous.
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Example 6
Any polynomial is continuous everywhere, that is , it is continuous on ℝ.
Any rational function is continuous wherever it is defined, that is, it is
continuous on its domain.
Example 7
Show that 𝑓 𝑥 = 𝑥 is continuous on ℝ = −∞, ∞ .
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Continuity of Compositions of Functions
If 𝑓 is continuous at 𝑐 and 𝑔 is continuous at 𝑓 𝑐 , then the composite
function 𝑔 ∘ 𝑓 given by (𝑔 ∘ 𝑓)(𝑥) = 𝑔 𝑓 𝑥 is continuous at 𝑐.
at is continuous at 𝑎.
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Example 8
Show that the following functions are continuous on their natural domains.
𝑥 2/3 𝑥−2
𝑏 𝑦= 𝑐 𝑦= 2
1 + 𝑥4 𝑥 −2
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Exercise 49
For what value of the constant 𝑎 is the function 𝑓 continuous everywhere.
𝑥 2 − 1, 𝑥<3
𝑓 𝑥 =ቊ
2𝑎𝑥, 𝑥≥3
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HOMEWORK
3, 5, 13, 25, 29, 50, 51
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