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