Continuity Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 / - - PowerPoint PPT Presentation

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Jul 19, 2023 119 likes •234 views

Continuity Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 / 10 Section 3.2 :: Continuity 2 / 10 Continuity at x = c A function f is continuous at x = c if the following three conditions are all satisfied: 1 f ( c ) is defined, 2 lim x

SLIDE 1

Continuity

Michael Freeze

MAT 151 UNC Wilmington

Summer 2013

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SLIDE 2

Section 3.2 :: Continuity

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SLIDE 3

Continuity at x = c

A function f is continuous at x = c if the following three conditions are all satisfied:

1 f (c) is defined, 2 lim

x→c f (x) exists, and

3 lim

x→c = f (c).

If f is not continuous at c, then we say it is discontinuous there.

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SLIDE 4

Identifying Discontinuities

−3 −2 −1 1 2 3 −4 −2 2 4 6

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SLIDE 5

Identifying Discontinuities

−8 −6 −4 −2 2 4 6 8 −8 −6 −4 −2 2 4 6 8 10

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SLIDE 6

Identifying Discontinuities

−8 −6 −4 −2 2 4 6 8 −6 −4 −2 2 4 6

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SLIDE 7

Types of Functions and their Continuity Properties

Polynomial Functions

continuous everywhere

Rational Functions

continuous wherever defined

Square Root Functions

continuous where radicand is non-negative

Exponential Functions

continuous everywhere

Logarithmic Functions

continuous on interval of positive real numbers

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SLIDE 8

Continuity on a Closed Interval

A function is continuous on a closed interval [a, b] if

1 it is continuous on the open interval (a, b), 2 it is continuous from the right at x = a, and 3 it is continuous from the left at x = b.

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SLIDE 9

Continuity of Piecewise-Defined Functions

Consider the function f (x) =      x − 1, x < 1 0, 1 ≤ x ≤ 4 x − 2, x > 4. Where is f (x) continuous?

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SLIDE 10

Continuity of Piecewise-Defined Functions

Consider the function f (x) =

x3 + k,

x ≤ 3 kx − 5, x > 3. Find the value of k so that f (x) is continuous at x = 3.

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