Lecture 5
Chapter 1 , Section 1.5
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Section 1.5: Definition of Continuity
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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.
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Each function drown in Figure 1.5.1 illustrates a discontinuity at 𝑥 = 𝑐
Figure 1.5.1 Doaa Al-Saafin
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Continuity on an Interval
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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.
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Some Properties of Continuous Functions
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Continuity of Polynomials and Rational Functions
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Continuity of Compositions
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Note:
1) lim𝑥→𝑐
|𝑔 𝑥 | = lim𝑥→𝑐
𝑔 𝑥
2) The absolute value of a continuous function is continuous.
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Section 1.6: Continuity of Trigonometric Functions
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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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1.6.2
1.6.3
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Solution (a):
Example 2
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Solution (b):
Example 2
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Solution (c):
Example 2
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