5.4 Continuity of a Function5.4 Continuity of a Function

Continuity of a Function

5.4 Continuity of a Function5.4 Continuity of a Function

A function is continuous if you can draw it without lifting your pen from the paper.

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Continuous

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Discontinuous

5.4 Continuity of a Function5.4 Continuity of a Function

A function is continuous at a point x = a if• f (a) is defined

• exists and

•

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limx→a

f (x)

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limx→a

f (x) = f (a)

5.4 Continuity of a Function5.4 Continuity of a Function

Some Important Continuous Functions • All polynomial functions• Rational functions, as long as the denominator

is nonzero• The sum, difference, product, and quotient (as

long as the denominator is nonzero) of two continuous functions

5.4 Continuity of a Function5.4 Continuity of a Function

2.a Determine whether each function is continuous or discontinuous. If discontinuous, state where it is discontinuous.

a) b)

a)

a)

a)

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f (x) = 5x − 7

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f (x) = x+1x−1

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f (x) = 125x 3 −5x

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f (x) = x+2x 4 −3x 3 −4x 2

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f (x) = 4x 3 − 5x 2 + 2x − 4

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Continuous

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Continuous

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Discontinuous atx =1

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Discontinuous atx = 0,−1,1

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Discontinuous atx = 0,−1,4