lecture 5 chapter 1 , section 1 -...
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Lecture 5
Chapter 1 , Section 1.5
Doaa Al-Saafin
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Section 1.5: Definition of Continuity
Doaa Al-Saafin
![Page 3: Lecture 5 Chapter 1 , Section 1 - ucarecdn.comucarecdn.com/4c939c20-725c-4b6c-8bf1-7a6fe6b936f5/Lecture 5.pdfLecture 5 Chapter 1 , Section 1.5 Doaa Al-Saafin. Section 1.5: Definition](https://reader034.vdocuments.us/reader034/viewer/2022043012/5fab6a6d7edd4c1c2b4fe250/html5/thumbnails/3.jpg)
Note: If one or more of the conditions of this definition fails to hold, then we will say
that 𝑓 has a discontinuity at 𝒙 = 𝒄.
Doaa Al-Saafin
![Page 4: Lecture 5 Chapter 1 , Section 1 - ucarecdn.comucarecdn.com/4c939c20-725c-4b6c-8bf1-7a6fe6b936f5/Lecture 5.pdfLecture 5 Chapter 1 , Section 1.5 Doaa Al-Saafin. Section 1.5: Definition](https://reader034.vdocuments.us/reader034/viewer/2022043012/5fab6a6d7edd4c1c2b4fe250/html5/thumbnails/4.jpg)
Each function drown in Figure 1.5.1 illustrates a discontinuity at 𝑥 = 𝑐
Figure 1.5.1 Doaa Al-Saafin
![Page 5: Lecture 5 Chapter 1 , Section 1 - ucarecdn.comucarecdn.com/4c939c20-725c-4b6c-8bf1-7a6fe6b936f5/Lecture 5.pdfLecture 5 Chapter 1 , Section 1.5 Doaa Al-Saafin. Section 1.5: Definition](https://reader034.vdocuments.us/reader034/viewer/2022043012/5fab6a6d7edd4c1c2b4fe250/html5/thumbnails/5.jpg)
Doaa Al-Saafin
![Page 6: Lecture 5 Chapter 1 , Section 1 - ucarecdn.comucarecdn.com/4c939c20-725c-4b6c-8bf1-7a6fe6b936f5/Lecture 5.pdfLecture 5 Chapter 1 , Section 1.5 Doaa Al-Saafin. Section 1.5: Definition](https://reader034.vdocuments.us/reader034/viewer/2022043012/5fab6a6d7edd4c1c2b4fe250/html5/thumbnails/6.jpg)
Continuity on an Interval
Doaa Al-Saafin
![Page 7: Lecture 5 Chapter 1 , Section 1 - ucarecdn.comucarecdn.com/4c939c20-725c-4b6c-8bf1-7a6fe6b936f5/Lecture 5.pdfLecture 5 Chapter 1 , Section 1.5 Doaa Al-Saafin. Section 1.5: Definition](https://reader034.vdocuments.us/reader034/viewer/2022043012/5fab6a6d7edd4c1c2b4fe250/html5/thumbnails/7.jpg)
In general, we will say a function 𝑓 is continuous from the left at 𝒄 if
lim𝑥→𝑐−
𝑓 𝑥 = 𝑓(𝑐)
and is continuous from the right at c if
lim𝑥→𝑐+
𝑓 𝑥 = 𝑓(𝑐)
Using this terminology we define continuity on a closed interval as follows.
Doaa Al-Saafin
![Page 8: Lecture 5 Chapter 1 , Section 1 - ucarecdn.comucarecdn.com/4c939c20-725c-4b6c-8bf1-7a6fe6b936f5/Lecture 5.pdfLecture 5 Chapter 1 , Section 1.5 Doaa Al-Saafin. Section 1.5: Definition](https://reader034.vdocuments.us/reader034/viewer/2022043012/5fab6a6d7edd4c1c2b4fe250/html5/thumbnails/8.jpg)
Doaa Al-Saafin
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Some Properties of Continuous Functions
Doaa Al-Saafin
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Continuity of Polynomials and Rational Functions
Doaa Al-Saafin
![Page 11: Lecture 5 Chapter 1 , Section 1 - ucarecdn.comucarecdn.com/4c939c20-725c-4b6c-8bf1-7a6fe6b936f5/Lecture 5.pdfLecture 5 Chapter 1 , Section 1.5 Doaa Al-Saafin. Section 1.5: Definition](https://reader034.vdocuments.us/reader034/viewer/2022043012/5fab6a6d7edd4c1c2b4fe250/html5/thumbnails/11.jpg)
Doaa Al-Saafin
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Doaa Al-Saafin
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Continuity of Compositions
Doaa Al-Saafin
![Page 14: Lecture 5 Chapter 1 , Section 1 - ucarecdn.comucarecdn.com/4c939c20-725c-4b6c-8bf1-7a6fe6b936f5/Lecture 5.pdfLecture 5 Chapter 1 , Section 1.5 Doaa Al-Saafin. Section 1.5: Definition](https://reader034.vdocuments.us/reader034/viewer/2022043012/5fab6a6d7edd4c1c2b4fe250/html5/thumbnails/14.jpg)
Doaa Al-Saafin
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Note:
1) lim𝑥→𝑐
|𝑔 𝑥 | = lim𝑥→𝑐
𝑔 𝑥
2) The absolute value of a continuous function is continuous.
Doaa Al-Saafin
![Page 16: Lecture 5 Chapter 1 , Section 1 - ucarecdn.comucarecdn.com/4c939c20-725c-4b6c-8bf1-7a6fe6b936f5/Lecture 5.pdfLecture 5 Chapter 1 , Section 1.5 Doaa Al-Saafin. Section 1.5: Definition](https://reader034.vdocuments.us/reader034/viewer/2022043012/5fab6a6d7edd4c1c2b4fe250/html5/thumbnails/16.jpg)
Doaa Al-Saafin
Section 1.6: Continuity of Trigonometric Functions
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Doaa Al-Saafin
Theorem 1.6.1 implies that the six basic trigonometric functions are continuous
on their domains. In particular, sin 𝒙 and cos 𝒙 are continuous everywhere.
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Doaa Al-Saafin
1.6.2
1.6.3
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Doaa Al-Saafin
Solution (a):
Example 2
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Doaa Al-Saafin
Solution (b):
Example 2
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Doaa Al-Saafin
Solution (c):
Example 2