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The Definition of Continuity

A function is continuous at a point x=c if the following three conditions are met...

The function is defined when x=c.

The Concept

Graph Examples

Notation

The function has to be defined when x=c, if there isn't a point for example, a hole then it is discontinuous.

Equation Examples

Graph Examples

Equation Examples

Jump- PW

Hole- Rational

Vertical Asymptote- Rational

PW fx-

...., x<5

...., x>5

If theres no equal sign on the inequalitites the fx is not defined

Rational fx-

The denominatore is 0. The numerator doesn't matter

Proper Notation

function defined f(c) is defined..

f(a) is defined

The limit as x approaches c of f(x) exists.

The Concept

Replace this text with a plain English explanation of what this condition means.

Notation

Graph Examples

Equation Examples

Graph Examples

In this space, upload some hand-drawn pictures of graphs where the limit does not exist and where it does exist. Be sure to point out where that is happening. Also show some where the function is defined and some where the function is not defined. Delete this text when you're done.

Equation Examples

Examples where the limit doesnt exist jump- piecewise f(x) asymptote- rational f(x)

PW f(x)- the 2 pieces dont equal each other at that point

Rational f(x)-

0/0- hole

#/0- vertical asymptote

Proper Notation

limit exists, lim f(x)= lim f(c).

limf(x) exist

x--a

The limit of f(x) as x approaches c is equal to f(c).

The Concept

the limit at f(x) nd c must be equal at the same y value.

Notation

Graph Examples

Equation Examples

Graph Examples

Equation Examples

We're looking for a hole-> rational fx by 0/0 with a dot-> piecewise fx

underneath

Proper Notation

f(c) =lim f(x)

x--c

lim f(x)= f(a)

x--a

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