Continuity Theorems

If f and g are continuous at a and c is a constant, then the following are also continuous

f + gf – gcffgf/g if g≠0

Any polynomial is continuous everywhere: it is continuous from (-∞,∞)

Any rational function is continuous wherever it is defined; in other words it is continuous on its domain

Example

Where is the following continuous?

3 22 11) ( )

5 3

x xf x

x

2) ( ) tanf x x

The following functions are continuous everywhere on their domains

PolynomialsRational functionsRoot functionsTrig functionsInverse trig functionsExponential functionsLogarithmic functions

Example

Where is the following continuous

2

ln sin( )

1

x xf x

x

Theorem

If f is continuous at b and then

or

lim ( )x ag x b

lim ( ( )) ( )x a

f g x f b

lim ( ( )) (lim ( ))x a x a

f g x f g x

Example:

Find

1

1limsin

1x

x

x

Theorem

If f is continuous, then

lim ( ) lim ( )n nx a x a

g x g x

Theorem

If g is continuous at a and f is continuous at g(a), then the composite function

is continuous at a

f g

Example

Where are the following continuous

21) ( ) sinh x x

2) ( ) ln(1 cos )h x x