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1.6 Continuity

Assigned work: pg 51 #4adef, bcdf,7,8,10-13A continuous curve is a curve without breaks, holes or jumps. Usually if we talk about a curve being discontinuous it is at a specific point. These are discontinuous…. HOLE JUMP ASYMPTOTE

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Slide21.6 Continuity

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Conditions for Continuity:

must exist

must exist (i.e. LHL=RHL) (i.e. condition 1 = condition 2)Note: For all polynomials so all polynomials are continuous.

Slide31.6 Continuity

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In the following examples state which condition fails:

cond

1 cond 2 cond 1 cond 3

Slide41.6 Continuity

When determining if a function is continuous you need to know where you should look for discontinuities…..For Rational functions: Look at when the denominator equals 0. If discontinuous, would fail condition 1 and 2If discontinuous, would be a hole or asymptote

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Slide51.6 Continuity

For Piecewise Functions:Look at the extreme domain valuesIf discontinuous, could fail condition 1, 2, and/or 3.If discontinuous, would be a jump or hole.NOTE: We are fussy about format for continuity so show ALL conditions in your work when showing if function is continuous or not.

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Slide61.6 Continuity

Rational Function Examples:Ex. 1:For what values of x is f(x) discontinuous. Show why, state which condition fails and what discontinuity you have.a)

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Hole at x = -1

Asymptote at x = 2We need to show conditions where denominator is = 0 (see next slide)

Slide71.6 Continuity

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So f(x) is discontinuous So f(x) is discontinuous

at x = -1 (hole at -1,-1/3) at x = 2 (vert. asymptote)

Slide81.6 Continuity

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So f(x) is discontinuous at x = 5

(

vert. asymptote

)

b)

Asymptote at x = 5

Now show conditions

Slide91.6 Continuity

Piecewise Function Examples:Ex. 2:For what values of x is f(x) discontinuous. Show why, state which condition fails and what discontinuity you have.a)

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Ask yourself where might f(x) be discontinuous? Look at domains – notice 1 & -1.

Slide101.6 Continuity

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So f(x) is continuous So f(x) is discontinuous

at x = -1 at x = 1 (jump)

Slide111.6 Continuity

b)

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Ask yourself where might f(x) be discontinuous? Look at

domains i.e. 0

So

f(x) is discontinuous at x= 0 (hole at 0,1)

Now Graph it

## 1.6 Continuity

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