Continuity!!Continuity!!

ca b

ca b

ca b

DefinitionsDefinitions

Continuity at a pointContinuity at a point: A function : A function ff is continuous at is continuous at cc if the following if the following three conditions are met:three conditions are met: is definedis defined existsexists

( )f c

lim ( )x c

f x

lim ( ) ( )x c

f x f c

DefinitionsDefinitions

Continuity on an open interval: Continuity on an open interval: A A function is continuous on an open function is continuous on an open interval if it is continuous at interval if it is continuous at each point in the intervaleach point in the interval

A function that is continuous on the A function that is continuous on the entire real line is everywhere entire real line is everywhere continuouscontinuous

( , )a b

( , )

DefinitionsDefinitions

A function is continuous on a A function is continuous on a closed interval if it is closed interval if it is continuous on the open interval continuous on the open interval and and

andand

lim ( ) ( )x a

f x f a

,a b

( , )a b

lim ( ) ( )x b

f x f b

c

y=f(x)

y

x

f(c)

ContinuousContinuous

c

y=f(x)

y

x

f(c)

Not continuousNot continuous

c

y=f(x)

y

x

NotNot continuouscontinuous

c

y=f(x)

y

x

Not continuousNot continuous

f(c)

c

y=f(x)

y

x

Not continuousNot continuous

c

y=f(x)

y

x

Not continuousNot continuous

The THREE requirements for The THREE requirements for a function to be continuous at a function to be continuous at

x=c …x=c …1.1. C must be in the domain of the function - you can C must be in the domain of the function - you can

find find f(c ), f(c ),

2.2. The right-hand limit must equal the left-hand limit which The right-hand limit must equal the left-hand limit which means that there is a LIMIT at x=c, and means that there is a LIMIT at x=c, and

3.3. AND…AND…

( ) lim ( )x c

f c f x

How does continuity relate How does continuity relate to an Etch-A-Sketch?to an Etch-A-Sketch?

If a function is If a function is continuous, there are continuous, there are

NO holes, NO holes, NO jumps, and NO jumps, and NO vertical NO vertical

asymptotes.asymptotes.

So, if you draw a continuous function, you should be able to draw it without lifting up your pencil….

just like an Etch A SketchEtch A Sketch!

Types of Discontinuities…Types of Discontinuities… Removable DiscontinuitiesRemovable Discontinuities – – cancan be be

“repaired”“repaired”– HoleHole (factor can be “factored out” of the (factor can be “factored out” of the

denominator)denominator) Essential DiscontinuitiesEssential Discontinuities – – cannotcannot be be

“repaired”“repaired”– JumpsJumps (usually found in piecewise (usually found in piecewise

functions)functions)– AsymptotesAsymptotes (can’t remove a (can’t remove a

factor/problem in the denominator) --- (like factor/problem in the denominator) --- (like 1/x) 1/x)

– Wildly oscillating functionsWildly oscillating functions – graph – graph (1/sin(x)) and keep zooming in !(1/sin(x)) and keep zooming in !