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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 Concept

the function hs to be defined when x=c, if there is no point, for example a hole, then it is discontinous

Equation Examples

The function is defined when x=c.

Graph Examples

Notation

Graph Examples

pwfx: {--,x<5

{--,x>5

Equation Examples

-Jump: --o o-- , piece wise

-Hole: --o-- , rational function

- Vertical asymptote: line go up , rational function

if theres no eqaul sign on the inequalities , the fx is not defined

Rational fx the denotminator is o numerator doesn't matter

Proper Notation

function defined f(c) is defined

f(a) is defined

The Concept

If the limit of f(x) as x approaches c is the same from both the right and the left, then we say that the limit of f(x) as x approaches c is L.

Equation Examples

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

Graph Examples

Notation

Graph Examples

Equation Examples

-Example where the limit doesnt exist jump asymptote

- Piece wise function(PWF): looking for the 2 pieces don't equal each other at that point

-Rational function: #/0

Proper Notation

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

lim f(x) exist

X --- a

The Concept

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

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

Equation Examples

Graph Examples

Notation

Graph Examples

Equation Examples

-We're looking for a hole with a dot underneath

-Rational function with 0/0

-Dot piece wise function

Proper Notation

f(c) = lim f(x)

x-> c

lim f(x) = f(a)

X --> a

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