ńņš. 17 |

t + (t)|t|n

t + Ā·Ā·Ā· +

f (t) = f (0) + f (0)t +

2! n!

where (t) ā’ 0 as t ā’ 0.

This exercise introduces several ideas which we use repeatedly in this

chapter so the reader should do it carefully.

Exercise 7.1.1. In this exercise we consider functions f, g : (ā’a, a) ā’ R

where a > 0.

(i) If f and g are diļ¬erentiable with f (t) ā¤ g (t) for all 0 ā¤ t < a and

f (0) = g(0), explain why f (t) ā¤ g(t) for all 0 ā¤ t < a.

(ii) If |f (t)| ā¤ |t|r for all t ā (ā’a, a) and f (0) = 0, show that |f (t)| ā¤

|t|r+1 /(r + 1) for all |t| < a.

141

142 A COMPANION TO ANALYSIS

(iii) If g is n times diļ¬erentiable with |g (n) (t)| ā¤ M for all t ā (ā’a, a)

and g(0) = g (0) = Ā· Ā· Ā· = g (nā’1) (0) = 0, show that

M |t|n

|g(t)| ā¤

n!

for all |t| < a.

(iv) If g is n times diļ¬erentiable in (ā’a, a) and g(0) = g (0) = Ā· Ā· Ā· =

g (n) (0) = 0, show, using (iii), that, if g (n) is continuous at 0, then

Ī·(t)|t|n

|g(t)| ā¤

n!

where Ī·(t) ā’ 0 as t ā’ 0.

(v) If f is n times diļ¬erentiable with |f (n) (t)| ā¤ M for all t ā (ā’a, a),

show that

nā’1

f (j) (0) j M |t|n

f (t) ā’ tā¤

j! n!

j=0

for all |t| < a.

(vi) If f is n times diļ¬erentiable in (ā’a, a), show that, if f (n) is contin-

uous at 0, then

n

f (j) (0) j Ī·(t)|t|n

f (t) ā’ tā¤

j! n!

j=0

where Ī·(t) ā’ 0 as t ā’ 0.

Restating parts (v) and (vi) of Exercise 7.1.1 we get two similar looking

but distinct theorems.

Theorem 7.1.2. (A global Taylorā™s theorem.) If f : (ā’a, a) ā’ R is n

times diļ¬erentiable with |f (n) (t)| ā¤ M for all t ā (ā’a, a), then

nā’1

f (j) (0) j M |t|n

f (t) ā’ tā¤ .

j! n!

j=0

Theorem 7.1.3. (The local Taylorā™s theorem). If f : (ā’a, a) ā’ R is n

times diļ¬erentiable and f (n) is continuous at 0, then

n

f (j) (0) j

t + (t)|t|n

f (t) =

j!

j=0

where (t) ā’ 0 as t ā’ 0.

143

Please send corrections however trivial to twk@dpmms.cam.ac.uk

We shall obtain other and more precise global Taylor theorems in the

course of the book (see Exercise K.49 and Theorem 8.3.20) but Theorem 7.1.2

is strong enough for the following typical applications.

Exercise 7.1.4. (i) Consider a diļ¬erentiable function e : R ā’ R which

obeys the diļ¬erential equation e (t) = e(t) with the initial condition e(0) = 1.

Quote a general theorem which tells you that, if a > 0, there exists an M

with |e(t)| ā¤ M for |t| ā¤ a. Show that

nā’1

tj M |t|n

e(t) ā’ ā¤

j! n!

j=0

for all |t| < a. Deduce that

nā’1

tj

ā’ e(t)

j!

j=0

as n ā’ ā, and so

ā

tj

e(t) =

j!

j=0

for all t.

(ii) Consider diļ¬erentiable functions s, c : R ā’ R which obey the diļ¬er-

ential equations s (t) = c(t), c (t) = ā’s(t) with the initial conditions s(0) = 0,

c(0) = 1. Show that

ā

(ā’1)j t2j+1

s(t) =

(2j + 1)!

j=0

for all t and obtain a similar result for c.

However, in this chapter we are interested in the local behaviour of func-

tions and therefoe in the local Taylor theorem. The distinction between local

and global Taylor expansion is made in the following very important example

of Cauchy.

Example 7.1.5. Consider the function F : R ā’ R deļ¬ned by

F (0) = 0

F (x) = exp(ā’1/x2 ) otherwise.

144 A COMPANION TO ANALYSIS

(i) Prove by induction, using the standard rules of diļ¬erentiation, that F

is inļ¬nitely diļ¬erentiable at all points x = 0 and that, at these points,

F (n) (x) = Pn (1/x) exp(ā’1/x2 )

where Pn is a polynomial which need not be found explicitly.

(ii) Explain why xā’1 Pn (1/x) exp(ā’1/x2 ) ā’ 0 as x ā’ 0.

(iii) Show by induction, using the deļ¬nition of diļ¬erentiation, that F is

inļ¬nitely diļ¬erentiable at 0 with F (n) (0) = 0 for all n. [Be careful to get this

part of the argument right.]

(iv) Show that

ā

F (j) (0) j

F (x) = x

j!

j=0

if and only if x = 0. (The reader may prefer to say that ā˜The Taylor expansion

of F is only valid at 0ā™.)

(v) Why does part (iv) not contradict the local Taylor theorem (Theo-

rem 7.1.3)?

[We give a diļ¬erent counterexample making use of uniform convergence in

Exercise K.226.]

Example 7.1.6. Show that, if we deļ¬ne E : R ā’ R by

if x ā¤ 0

E(x) = 0

E(x) = exp(ā’1/x2 ) otherwise,

then E is an inļ¬nitely diļ¬erentiable function with E(x) = 0 for x ā¤ 0 and

E(x) > 0 for x > 0

Cauchy gave his example to show that we cannot develop the calculus

algebraically but must use , Ī“ techniques. In later courses the reader will

see that his example encapsulates a key diļ¬erence between real and complex

analysis. If the reader perseveres further with mathematics she will also ļ¬nd

the function E playing a useful rĖle in distribution theory and diļ¬erential

o

geometry.

A simple example of the use of the local Taylor theorem is given by the

proof of (a version of) Lā™HĖpitalā™s rule in the next exercise.

o

Exercise 7.1.7. If f, g : (ā’a, a) ā’ R are n times diļ¬erentiable and

f (0) = f (0) = Ā· Ā· Ā· = f nā’1 (0) = g(0) = g (0) = Ā· Ā· Ā· = g (nā’1) (0) = 0

145

Please send corrections however trivial to twk@dpmms.cam.ac.uk

but g (n) (0) = 0 then, if f (n) and g (n) are continuous at 0, it follows that

f (n) (0)

f (t)

ā’ (n)

g(t) g (0)

as t ā’ 0.

It should be pointed out that the local Taylor theorems of this chapter

(and the global ones proved elsewhere) are deep results which depend on

the fundamental axiom. The fact that we use mean value theorems to prove

them is thus not surprising ā” we must use the fundamental axiom or results

derived from it in the proof.

(Most of my readers will be prepared to accept my word for the statements

made in the previous paragraph. Those who are not will need to work through

the next exercise. The others may skip it.)

Exercise 7.1.8. Explain why we can ļ¬nd a sequence of irrational numbers

an such that 4ā’nā’1 < an < 4ā’n . We write I0 = {x ā Q : x > a0 } and

In = {x ā Q : an < x < anā’1 }

[n = 1, 2, 3, . . . ]. Check that, if x ā In , then 4ā’nā’1 < x < 4ā’n+1 [n ā„ 1].

We deļ¬ne f : Q ā’ Q by f (0) = 0 and f (x) = 8ā’n if |x| ā In [n ā„ 0]. In

what follows we work in Q.

(i) Show that

f (h) ā’ f (0)

ā’0

h

as h ā’ 0. Conclude that f is diļ¬erentiable at 0 with f (0) = 0.

(ii) Explain why f is everywhere diļ¬erentiable with f (x) = 0 for all x.

Conclude that f is inļ¬nitely diļ¬erentiable with f (r) = 0 for all r ā„ 0.

(iii) Show that

f (h) ā’ f (0)

ā’ā

h2

as h ā’ 0. Conclude that, if we write

f (0) 2

h + (h)h2 ,

f (h) = f (0) + f (0)h +

2!

0 as h ā’ 0. Thus the local Taylor theorem (Theorem 7.1.3) is

then (h)

false for Q.

146 A COMPANION TO ANALYSIS

7.2 Some many dimensional local Taylor the-

orems

In the previous section we used mean value inequalities to investigate the

local behaviour of well behaved functions f : R ā’ R. We now use the same

ideas to investigate the local behaviour of well behaved functions f : Rn ā’ R.

It turns out that, once we understand what happens when n = 2, it is easy

to extend the results to general n and this will be left to the reader.

Here is our ļ¬rst example.

Lemma 7.2.1. We work in R2 and write 0 = (0, 0).

(i) Suppose Ī“ > 0, and that f : B(0, Ī“) ā’ R has partial derivatives f ,1

and f,2 with |f,1 (x, y)|, |f,2 (x, y)| ā¤ M for all (x, y) ā B(0, Ī“). If f (0, 0) = 0,

then

|f (x, y)| ā¤ 2M (x2 + y 2 )1/2

for all (x, y) ā B(0, Ī“).

(ii) Suppose Ī“ > 0, and that g : B(0, Ī“) ā’ R has partial derivatives

g,1 and g,2 in B(0, Ī“). Suppose that g,1 and g,2 are continuous at (0, 0) and

g(0, 0) = g,1 (0, 0) = g,2 (0, 0) = 0. Then writing

g((h, k)) = (h, k)(h2 + k 2 )1/2

we have (h, k) ā’ 0 as (h2 + k 2 )1/2 ā’ 0.

Proof. (i) Observe that the one dimensional mean value inequality applied

to the function t ā’ f (x, t) gives

|f (x, y) ā’ f (x, 0)| ā¤ M |y|

whenever (x, y) ā B(0, Ī“) and the same inequality applied to the function

s ā’ f (s, 0) gives

|f (x, 0) ā’ f (0, 0)| ā¤ M |x|

whenever (x, 0) ā B(0, Ī“). We now apply a taxicab argument (the idea

behind the name is that a New York taxicab which wishes to get from (0, 0)

to (x, y) will be forced by the grid pattern of streets to go from (0, 0) to (x, 0)

and thence to (x, y)) to obtain

|f (x, y)| = |f (x, y) ā’ f (0, 0)| = |(f (x, y) ā’ f (x, 0)) + (f (x, 0) ā’ f (0, 0))|

ā¤ |f (x, y) ā’ f (x, 0)| + |f (x, 0) ā’ f (0, 0)| ā¤ M |y| + M |x|

ā¤ 2M (x2 + y 2 )1/2

147

Please send corrections however trivial to twk@dpmms.cam.ac.uk

for all (x, y) ā B(0, Ī“).

(ii) Let > 0 be given. By the deļ¬nition of continuity, we can ļ¬nd a Ī“1 ( )

such that Ī“ > Ī“1 ( ) > 0 and

|g,1 (x, y)|, |g,2 (x, y)| ā¤ /2

for all (x, y) ā B(0, Ī“1 ( )). By part (i), this means that

|g(x, y)| ā¤ (x2 + y 2 )1/2

for all (x, y) ā B(0, Ī“1 ( )) and this gives the desired result.

Theorem 7.2.2. (Continuity of partial derivatives implies diļ¬eren-

tiability.) Suppose Ī“ > 0, x = (x, y) ā R2 , B(x, Ī“) ā E ā R2 and that

f : E ā’ R. If the partial derivatives f,1 and f,2 exist in B(x, Ī“) and are

continuous at x, then, writing

f (x + h, y + k) = f (x, y) + f,1 (x, y)h + f,2 (x, y)k + (h, k)(h2 + k 2 )1/2 ,

we have (h, k) ā’ 0 as (h2 + k 2 )1/2 ā’ 0. (In other words, f is diļ¬erentiable

at x.)

Proof. By translation, we may suppose that x = 0. Now set

g(x, y) = f (x, y) ā’ f (0, 0) ā’ f,1 (0, 0)x ā’ f,2 (0, 0)y.

We see that g satisļ¬es the hypotheses of part (ii) of Lemma 7.2.1. Thus g

satisļ¬es the conclusions of part (ii) of Lemma 7.2.1 and our theorem follows.

Although this is not one of the great theorems of all time, it occasionally

provides a useful short cut for proving functions diļ¬erentiable1 . The following

easy extensions are left to the reader.

Theorem 7.2.3. (i) Suppose Ī“ > 0, x ā Rm , B(x, Ī“) ā E ā Rm and that

f : E ā’ R. If the partial derivatives f,1 , f,2 , . . . f,m exist in B(x, Ī“) and are

continuous at x, then f is diļ¬erentiable at x.

(ii) Suppose Ī“ > 0, x ā Rm , B(x, Ī“) ā E ā Rm and that f : E ā’ Rp . If

the partial derivatives fi,j exist in B(x, Ī“) and are continuous at x [1 ā¤ i ā¤

p, 1 ā¤ j ā¤ m], then f is diļ¬erentiable at x.

1

I emphasise the word occasionally. Usually, results like the fact that the diļ¬erentiable

function of a diļ¬erentiable function is diļ¬erentiable give a faster and more satisfactory

proof.

148 A COMPANION TO ANALYSIS

Similar ideas to those used in the proof of Theorem 7.2.2 give our next

result which we shall therefore prove more expeditiously. We write

f,ij (x) = (f,j ),i (x),

or, in more familiar notation,

ā‚2f

f,ij = .

ā‚xi ā‚xj

Theorem 7.2.4. (Second order Taylor series.) Suppose Ī“ > 0, x =

(x, y) ā R2 , B(x, Ī“) ā E ā R2 and that f : E ā’ R. If the partial derivatives

f,1 , f,2 , f,11 , f,12 , f,22 exist in B(x, Ī“) and f,11 , f,12 , f,22 are continuous at x,

then writing

f ((x + h, y + k)) =f (x, y) + f,1 (x, y)h + f,2 (x, y)k

+ (f,11 (x, y)h2 + 2f,12 (x, y)hk + f,22 (x, y)k 2 )/2 + (h, k)(h2 + k 2 ),

we have (h, k) ā’ 0 as (h2 + k 2 )1/2 ā’ 0.

Proof. By translation, we may suppose that x = 0. By considering

f (h, k) ā’ f (0, 0) ā’ f,1 (0, 0)h ā’ f,2 (0, 0)k ā’ (f,11 (0, 0)h2 + 2f,12 (0, 0)hk + f,22 (0, 0)k 2 )/2,

we may suppose that

f (0, 0) = f,1 (0, 0) = f,2 (0, 0) = f,11 (0, 0) = f,12 (0, 0) = f,22 (0, 0).

If we do this, our task reduces to showing that

f (h, k)

ā’0

h2 + k 2

as (h2 + k 2 )1/2 ā’ 0.

To this end, observe that, if > 0, the continuity of the given partial

derivatives at (0, 0) tells us that we can ļ¬nd a Ī“1 ( ) such that Ī“ > Ī“1 ( ) > 0

and

|f,11 (h, k)|, |f,12 (h, k)|, |f,22 (h, k)| ā¤

for all (h, k) ā B(0, Ī“1 ( )). Using the mean value inequality in the manner

of Lemma 7.2.1, we have

|f,1 (h, k) ā’ f,1 (h, 0)| ā¤ |k|

149

Please send corrections however trivial to twk@dpmms.cam.ac.uk

and

|f,1 (h, 0) ā’ f,1 (0, 0)| ā¤ |h|

and a taxicab argument gives

|f,1 (h, k)| = |f,1 (h, k) ā’ f,1 (0, 0)| = |(f,1 (h, k) ā’ f,1 (h, 0)) + (f,1 (h, 0) ā’ f,1 (0, 0))|

ā¤ |f,1 (h, k) ā’ f,1 (h, 0)| + |f,1 (h, 0) ā’ f,1 (0, 0)| ā¤ (|k| + |h|)

for all (h, k) ā B(0, Ī“1 ( )). (Or we could have just applied Lemma 7.2.1 with

f replaced by f,1 .) The mean value inequality also gives

|f,2 (0, k)| = |f,2 (0, k) ā’ f,2 (0, 0)| ā¤ |k|.

Now, applying the taxicab argument again, using the mean value inequal-

ity and the estimates of the ļ¬rst paragraph, we get

|f (h, k)| = |f (h, k) ā’ f (0, 0)| = |(f (h, k) ā’ f (0, k)) + (f (0, k) ā’ f (0, 0))|

ā¤ |f (h, k) ā’ f (0, k)| + |f (0, k) ā’ f (0, 0)|

ā¤ sup |f,1 (sh, k)||h| + sup |f,2 (0, tk)||k|

0ā¤sā¤1 0ā¤tā¤1

ā¤ (|k| + |h|)|h| + |k|2

ā¤ 3 (h2 + k 2 ).

Since was arbitrary, the result follows.

Exercise 7.2.5. Set out the proof of Theorem 7.2.4 in the style of the proof

of Theorem 7.2.2.

We have the following important corollary.

Theorem 7.2.6. (Symmetry of the second partial derivatives.) Sup-

pose Ī“ > 0, x = (x, y) ā R2 , B(x, Ī“) ā E ā R2 and that f : E ā’ R.

If the partial derivatives f,1 , f,2 , f,11 , f,12 , f,21 f,22 exist in B(x, Ī“) and are

continuous at x, then f,12 (x) = f,21 (x).

Proof. By Theorem 7.2.4, we have

f (x + h, y + k) =f (x, y) + f,1 (x, y)h + f,2 (x, y)k

+ (f,11 (x, y)h2 + 2f,12 (x, y)hk + f,22 (x, y)k 2 )/2 + 2

+ k2)

1 (h, k)(h

with 1 (h, k) ā’ 0 as (h2 + k 2 )1/2 ā’ 0. But, interchanging the rĖle of ļ¬rst

o

and second variable, Theorem 7.2.4 also tells us that

f (x + h, y + k) =f (x, y) + f,1 (x, y)h + f,2 (x, y)k

+ (f,11 (x, y)h2 + 2f,21 (x, y)hk + f,22 (x, y)k 2 )/2 + 2

+ k2)

2 (h, k)(h

150 A COMPANION TO ANALYSIS

with 2 (h, k) ā’ 0 as (h2 + k 2 )1/2 ā’ 0.

Comparing the two Taylor expansions for f (x + h, y + k), we see that

2

+ k2) = 2

+ k2)

f,12 (x, y)hk ā’ f,21 (x, y)hk = ( 1 (h, k) ā’ 2 (h, k))(h 3 (h, k)(h

ā’ 0 as (h2 + k 2 )1/2 ā’ 0. Taking h = k and dividing by h2 we

with 3 (h, k)

have

f,12 (x, y) ā’ f,21 (x, y) = 2 3 (h, h) ā’ 0

as h ā’ 0, so f,12 (x, y) ā’ f,21 (x, y) = 0 as required.

ńņš. 17 |