x

1

y

1 1 2x

1

2

3

y

The Continuity of a Function at a Point

Definition: f is continuous at x = a 5 )()(lim afxfax

The graph of an everywhere continuous function, y = f ( x ) , possesses no breaks or jumps. It can be drawn without ever lifting your pencil off the page. If f is not continuous at x = a then it is said to be discontinuous there.

Classification of 3 Types Discontinuities at x = a

( i ) Removable: In case Lxfax

)(lim , where L 2 but f ( a ) ≠ L .

Example:

Let

10

11

1)(

2

xif

xifx

xxf

.

Then f has a removable discontinuity at x = 1 . To see this, observe that the formula for f simplifies to:

10

1)(

xif

xifxxf

1 .

so 211)1(lim)(lim

11

xxf

xx.

Thus, we could remove the discontinuity at x = 1 by redefining its value there to be 2. That is, the function

12

11

1)(

2

*

xif

xifx

xxf

is continuous at x = 1 .

( ii ) Jump: In case lim ( )

x a

f x L

, lim ( ) x a

f x M

where L, M 2 but L ≠ M .

Example:

Let

00

01)(

xif

xifxu

.

Here, 1

)1(lim)(lim

00 xxxu

but

0 0lim ( ) lim (0) 0

x xu x

.

r f has a jump discontinuity at x = 0. O

O

The Relation of Continuity to Differentiability

2

12 1 2x

2

4

y

( iii ) Infinite: In case either lim ( )x a

f x

or lim ( )x a

f x

is infinite valued .

Example: Let 2

10

( )2 0

if xf x x

if x

.

Observe that

0

11lim)(lim

200 x

xfxx

and

0

11lim)(lim

200 x

xfxx

r f has an infinite discontinuity at x = 0. Relation to Differentiability

Recall the definitions:

I. Geometrically, the graph of y = f ( x ) will not have a break ( i.e., a hole, jump, or vertical asymptote) at the point P = ( a , f ( a ) ) if f is continuous at x = a .

II.

Geometrically, the graph of y = f ( x ) will be smooth (i.e., it will not have a break or a sudden change in direction) at the point P = ( a , f ( a ) ) if f is differentiable at x = a .

Thus, differentiability is a stronger, more restrictive condition to impose upon a function than is continuity.

Example: We have already seen that the unit step function, u ( x ) , has a jump discontinuity at x = 0 . We will now verify that u ( x ) is also not differentiable at x = 0 . The right-side derivative of u at x = 0 has value:

0

xxx

uxuu

xxx

0lim

11lim

)0()(lim)0(

000

f continuous at x = a )()(lim afxf

ax

5

f differentiable at x = a

Lx

afxafx

)()(lim

0 for some L 2 5

O

u ’+ ( 0 ) . =

2

10

( )2 0

if xf x x

if x

.

3 but the left-side derivative at x = 0 has value:

0

xxx

uxuu

xxx

1lim

1lim

)0()(lim)0(

000

Consequently,

x

uxuu

x

)0()(lim)0(

0 does not exist.

I.e., u ( x ) is not differentiable at x = 0 This example helps to motivate the theorem:

; Geometrically, if the graph of y = f ( x ) has a break at the point P = ( a , f ( a ) ) then this graph cannot have a definable slope, i.e. it is not smooth, at point P. The logically equivalent contra positive to this theorem is:

;; Geometrically, this says that if the graph of y = f ( x ) is smooth at the point P = ( a , f ( a ) ), i.e. it has slope f ’ ( a ) , then it cannot possess a break at point P.

It is useful at this point in our discussion to introduce some basic Rules of Logic.

Consider the propositions: p: “It is a sunny day.” q: “We will go to the park today.”

and the related conditional statements:

( i ) p 0 q (p implies q) “If it is a sunny day then we will go to the park today.” ( i i ) ¬ q 0 ¬ p (not q implies not p) “If we do not go to the park today then it is not a sunny day.” ( i i i ) q 0 p (q implies p) “If we go to the park today then it is a sunny day.”

Statement ( i i ) is known as the contra positive of ( i ) and it is logically equivalent to it, i.e. it has the same truth value.

Statement ( i i i ) is the converse of ( i ).

u ’ _ ( 0 ) = +N

u ’ + ( 0 ) = 0

f discontinuous at x = a 0

f not differentiable at x = a

f differentiable at x = a 0 f continuous

at x = a

O

u ’_( 0 ) =

u ’( 0 ) =

4 Observe that ( i ) and ( i i i ) are not equivalent in this case.

Statements ( i ) and ( i i ) both mean: “We always go to the park on sunny days.”

Statement ( i i i ) means: “We only go to the park on sunny days.” It is also the case that the converse of our theorem ;; is not true in general. That is,

Problem: Construct a function that is continuous at x = 0 but is not differentiable at x = 0 .

f continuous at x = a

f differentiable at x = a L