Continuity – Part 2Continuity – Part 2

The THREE requirements for a The THREE requirements for a function to be continuous at x=c function to be continuous at x=c ……

1.1. C must be in the domain of the function - C must be in the domain of the function - you can find you can find f(c ), f(c ),

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

3.3. ANDAND( ) lim ( )

x cf c f x

Properties of ContinuityProperties of Continuity

If If bb is a real number and is a real number and ff and and gg are continuous are continuous at , then the following functions are at , then the following functions are already continuous at already continuous at c…c…

1. 1.

Recall:Recall:

So ifSo if

thenthen i.e., is i.e., is continuouscontinuous

lim ( ) lim ( )x c x c

bf x b f x

x c

bf

lim ( ) ( )x c

f x f c

lim ( ) lim ( ) ( )x c x c

bf x b f x bf c

bf

Properties of ContinuityProperties of ContinuityIf If bb is a real number and is a real number and ff and and gg are are

continuous at , then the following continuous at , then the following functions are already continuous at functions are already continuous at c…c…

2. 2. Recall:Recall: So ifSo ifthenthen i.e., is continuousi.e., is continuous

lim ( ) ( ) lim ( ) lim ( )x c x c x c

f x g x f x g x

x c

f g

lim ( ) ( ) and lim ( ) ( )x c x c

f x f c g x g c

lim ( ) ( ) lim ( ) lim ( ) ( ) ( )

x c x c x cf x g x f x g x f c g c

f g

Properties of ContinuityProperties of ContinuityIf If bb is a real number and is a real number and ff and and gg are are

continuous at , then the following continuous at , then the following functions are also continuous at functions are also continuous at c…c…

3. 3. Recall:Recall: So ifSo ifthenthen i.e., i.e., is continuousis continuous

lim ( ) ( ) lim ( ) lim ( )x c x c x c

f x g x f x g x

x c

fg

lim ( ) ( ) and lim ( ) ( )x c x c

f x f c g x g c

lim ( ) ( ) lim ( ) lim ( ) ( ) ( )x c x c x c

f x g x f x g x f c g c

fg

Properties of ContinuityProperties of ContinuityIf bIf b is a real number and is a real number and ff and and gg are continuous are continuous

at , then the following functions are also at , then the following functions are also continuous at continuous at c…c…

4.4.

Recall:Recall:

So ifSo if

thenthen i.e., i.e., i is s continuouscontinuous

lim ( )( )lim

( ) lim ( )x c

x cx c

f xf x

g x g x

x c

f

g

lim ( ) ( ) and lim ( ) ( )x c x c

f x f c g x g c

lim ( )( ) ( )

lim( ) lim ( ) ( )

x c

x cx c

f xf x f c

g x g x g c

f

g

Properties of ContinuityProperties of ContinuityIf If gg is continuous at is continuous at c c and and ff is continuous at is continuous at g(c)g(c), ,

then the composite function ( then the composite function ( ff ◦ ◦ gg)()(xx) is) is also also continuous at continuous at c.c.

Recall:Recall:

So ifSo if

thenthen i.e., i.e., ( ( ff ◦ ◦ gg)()(xx) i) is continuouss continuous

lim ( ( )) (lim ( ))x c x c

f g x f g x

( )lim ( ) ( ) and lim ( ) ( ( ))x c x g c

g x g c f x f g c

lim ( ( )) (lim ( )) ( ( ))x c x c

f g x f g x f g c

Determine whether the function is continuous at (a) x = -1 (b) x = 2.

(REQ#1) f (-1) = 2(-1) + 3 = 1

f (x) is CONTINUOUS at x = -1.

Is the function continuous at -1?

(REQ#2)

1So, lim ( ) 1

xf x

1lim ( )1) 1(x

xf f

(REQ#3)

f (2) = -2 + 5 = 3

f (x) is NOT CONTINUOUS at x = 2.

Is f (x) continuous at x = 2?

(REQ#3)

(REQ#3)

(REQ#2)

There is NO LIMIT

(no need to test – does not meet requirement #2!)

• What does the Graph look like? What does the Graph look like?

• Quickly graph this piecewise function and see if it confirms Quickly graph this piecewise function and see if it confirms our conclusions!our conclusions!

This is called

a JUMP discontinuity!

Polynomial Functions

Every polynomial function is continuous at every

real number.

Functions that are continuous in their respective domains…

Rational Functions {x: q(x) ≠ 0}

Power Functions

if n is odd: all real #sif n is even: {x: x > 0}

( )( )

( )

p xR x

q x

( ) nf x x

Functions that are continuous in their respective domains…

Logarithmic Functions{x: x > 0}

Exponential FunctionsAll Real Numbers

Functions that are continuous in their respective domains…

Sin(x) or Cos(x)

All real numbers

Tan(x) or Sec(x)

Cot(x) or Csc(x)

: , any integer2

n n

: , any integern n

Functions that are continuous in their respective domains…

Arccos(x) or Arcsin(x)

-1 ≤ x ≤ 1

Arctan(x) or arccot(x)

All real numbers

Arcsec(x) or Arccsc (x) |x| ≥ 1

Functions and Limits to KNOW!

function

Left-hand LIMIT

Output

f(a)

Right-hand LIMIT

LIMIT Cont at

X = a?

Type of discont.

?

Remov.

Or Nonre.?

x/x 1 Undef. 1 1 No Hole Remov.

Sin(x)/x 1 Undef. 1 1 No Hole Remov.

|x|/x -1 Undef. 1 DNE No Jump Nonre.

1/x -∞ Undef. ∞ DNE No Vert.

Asymp.

Nonre.

1/x2 ∞ Undef. ∞ ∞ No Vert.

Asymp.

Nonre.

Sin(1/x) DNE Undef. DNE DNE No Wild

Ocillat,

Nonre.