Math 31A October 8 Lesson Plan

• Warm-up problem: Consider the function

{ET, ×-4

fix) = 10 , ×-4.

Which of the following best describes the

behavior of f

at ✗ = 4?

A. There is a removable discontinuity. *

B. There is a jump discontinuity.

C. There is an infinite discontinuity.

D- There is an oscillating discontinuity.

E. The function is continuous.

To see why the correct answer is A,

note that we can

factor ✗ 2 -16 = (✗ -4) Cx + 4)

(this is a

difference of squares). So =

X + 4 for

all ✗ #4. Thus, ¥54T Lex =

¥34-Lex> = 4+4=8. However, f (4)

=L 0 #8. So f has a removable

discontinuity at ✗ = 4.

I. Review

• In lecture, you've been discussing

limit laws and continuity

(including the various types of discontinuities].

• Your homework is on evaluating limits

and applying the

limit laws.

• Are there any questions on this

material?

Spend as much time as needed on questions.

I. Problems

(2.4. 6 6) Find the value of the constant c

that makes the function continuous at the

point × =3.

{2x + 9×-1 , for

× =3

fix> =-4x + c , for

✗ > 3.

Sot To check that it is possible to choose

such a value

for c, we must first check that the left-

and right -

hand limits of f at ✗ =3 "For all ✗

<3, we have f (x) = 2x + 9×-1, so

3- fix> = 3-

(2×+9×-1)

= 2 ¥33-✗ + 9

Its-I'

= 2. 3 + 9 - 1/3

= 6 + 3 = 9.

For all ✗ > 3, we have f (x) = -4x

+ e, so

¥3-Aca- ¥3T c- 4×+0

=-4 ⇒ + × +

¥3-c

= -4 -3 + c

= -12 + C.

Now, in order for ¥33 LID to exist,

we need the

left- and right- hand limits to be

equal. So we

need to solve for a such that

9 = -12 + c.

Thus c= 21.

• The floor function is defined as follows:

For

each ×, fix) is the greatest integer less

than or equal to ×. CE. g. , fL3-27)

=3, I C- 1.73=-2.) where do the

discontinuities of this function occur?

Describe them.

Sot The floor function looks like this i

There is a jump discontinuity at each

integer. If n is an integer>

IF-LCD = n-1, while ✗ Ent

fo-n.

•2- 4.97" Show" that f is a

discontinuous

function for all ×, where LCD is

defined as follows:

{is ✗ is rational

fix) = -1, × is irrational

Show that It is continuous for all

✗ .

Sot Since 12 is just the constant

function 1> it is continuous

everywhere.

To see that f is not continuous

at any ✗ , let ✗ be any real

number. Between any two distinct reel

numbers, there exist both a rational

number and an irrational number. Thus, no

matter how close we get to ✗ , we will

meet values ×, and ×, such that

Lex,> = 1 and fly)=-1. So, in fact,

neither the left- no-the right-hand

limit at ✗ exists. (This is a bit

informal. To improve it, we could do

what's called an

epsilon -delta proof, but proofs are

beyond the scope of this course?

• 2.4.71 Draw the graph of a function on

[95] with

the given properties: f is not continuous at ✗

=L, but

✗ "It fix> and I - fix> exist and

are equal.

so We are being asked to draw a graph of

a function with a removable discontinuity at ✗

= 1. There are many possibilities,

one of which is sketched below.

'2.4.71Draw the graph of a function on

[95] with

the given properties: if has a removable

discontinuity at

✗ =L, a jump discontinuity at ✗ =L, and

¥3-Lex> =-• ,

33+11×3=2.

⇒ Again> there are many possibilities. Recall

that we have a:

• removable discontinuity when the left-

and right - hand

limits exist and are equal;

• jump discontinuity when the left- and

right-hand limits exist but are unequal.

•2.4.90: which of the following quantities

would be represented by continuous

functions of time and which would have one

or more discontinuities?

⑨ Velocity of an airplane during a flight

⑥ Temperature in a room under ordinary

conditions

⑨ Value of a bank account with interest

paid yearly

② salary of a teacher

② Population of the world

So Some of these are a little debatable,

but they are fun to think about!

⑨ Continuous. Even in cases of rapid

acceleration or deceleration,

the velocity does not change drastically in a

single instant.

⑥ continuous. We usually conceive of

temperature as

changing gradually-

⑨ Jump discontinuities. At the instant when

the interest is

paid, the value of the account jumps

from one point

to another.

② Jump discontinuities. At the instant

when the salary is

raised, the payment value jumps from

one point to

another.

⑨ Jump discontinuities. At the instant

when someone is born

or dies, the population jumps or falls by

a whole

number.

Graph these if students wish to see

them.

22.4.81 Suppose that f and g are

discontinuous at

✗ =L. Does it follow that ft ng is

discontinuous at ✗ =L?

If not, give a counterexample-

Sot This does not follow in general.

To try and

find a counterexample, we'll consider

additive

cancelations. (This is always a natural

place to look when we're dealing with

addition!) Let's consider

{1. ✗ so

Lex> = -1, ✗ <0 and

{ 'is ✗ so

go = Is ✗ <0.

Notice that, for all ✗ , LCD +

go =D. So Ltg is continuous

everywhere, and in particular at

✗ =D. However, each of f and ng hes

a jump discontinuity

at ✗ =D. So this is a valid

counterexample. (There are many more

counterexamples?