1.2
1
Continuous Functions in R
1.3
This terminology and notation is not widely used.
1.6 We shall denote the domain of a function f by dom( f ).
1.1 We begin by recalling the definition of a convergent
sequence of real numbers.
1.7 Suppose that A is a non-empty subset of R and that (an )
is a sequence such that, for all n ∈ N, an ∈ A. Then we say that
(an ) is a sequence in A and we write (an ) ⊂ A.
√
For example, (1/n) ⊂ [0, 1] and ((−1)n / n) ⊂ [−1, 1].
1.2 Definition
We say that the sequence (an ) converges to the limit a ∈ R,
and we write
lim(an ) = a,
n
if and only if, ∀ε > 0, ∃N ∈ N such that
∀n ∈ N, if n > N then |an − a| < ε.
1.3 Sometimes we write N(ε) or Nε instead of N to emphasis
that the value of N depends on the value of ε.
1.4 Next, we introduce a useful piece of notation.
1.5 Suppose that A and B are non-empty sets. We say that f is
a function out of A into B and we write
f : A B.
if and only if there is a non-empty set D ⊂ A such that
f : D → B. In other words f : A B if and only if the domain
of f is a subset of A.
For example, it is not correct to write
√
f : R → R : x 7→ x
1.8 Definition
Suppose that f : R R and that a ∈ dom( f ). Then we say
that f is continuous at a if and only if, for every sequence (an )
in dom( f ) that converges to a, the sequence ( f (an )) converges
to f (a).
1.9 The first rigorous definition of continuity is due to
Augustin-Louis Cauchy (1789–1857). We shall prove below
that Cauchy’s definition is equivalent to 1.8.
1.10 Definition
Suppose that f : R R and that A is a non-empty subset of
dom( f ). Then we say that f is continuous on A if and only if,
for all a ∈ A, f is continuous at a.
If f is continuous on dom( f ) then we say that f is a
continuous function or simply that f is continuous.
1.11 Several important theorems about continuous functions
can be proved very easily by using standard theorems on
convergent sequences.
because f (x) is defined only if x ≥ 0, but it is correct to write
√
f : R R : x 7→ x.
1.4
1.12 Theorem
Suppose that λ ∈ R and that
1.5
1.16 Corollary
For all n ∈ N, suppose that
pn : R → R : x 7→ xn .
f : R → R : x 7→ λ .
Then, for all n ∈ N, pn is continuous (on R).
Then f is continuous (on R). In other words, every constant
function is a continuous function.
Proof Suppose that a ∈ R and that (an ) is a sequence that
converges to a. For all n ∈ N, f (an ) = λ . Therefore
lim( f (an )) = lim(λ ) = λ = f (a).
n
1.17 Definition
Suppose that n ∈ Z+ and that, for all k = 0, 1, . . . , n, ak ∈ R and
that an 6= 0. Then the function f : R → R defined by
(1)
n
Therefore f is continuous at a. Therefore f is continuous.
n
f (x) =
∑ ak xk = a0 + a1 x + a2 x2 + · · · + an xn
k=0
1.13 Theorem
Suppose that f : R → R : x 7→ x. Then f is continuous. In other
words, the identity function is continuous. is called a polynomial function of degree n.
1.18 Example
The function
1.14 In the rest of this chapter, when we use a phrase such as
“ f is continuous at a” we shall take it to imply that f : R R
and that a ∈ dom( f ).
is a polynomial function of degree 3.
1.15 Theorem
Suppose that f and g are both continuous at a. Then
1.19 Theorem
Every polynomial function is continuous.
(i) f + g is continuous at a.
(ii) f · g is continuous at a.
(iii) If g(a) 6= 0 then f /g is continuous at a.
f : x → 5x3 + 2x + 1 (= 1 + 2x + 0x2 + 5x3 )
1.20 Definition
A function f : R R is said to be a rational function if and
only if f = p/q where p and q are polynomial functions; the
domain of f is the set { x ∈ R | q(x) 6= 0 }.
1.21 Theorem
Every rational function is continuous.
1.6
1.22 When we say that a rational function f is continuous we
do not mean that it is continuous on R but that it is continuous
on its domain. For example, if
(x + 1)4
x(x − 1)2
f : x 7→
1.7
1.27 Theorem (The Intermediate Value Theorem, IVT)
Suppose that a, b ∈ R, that a < b, and that f : R R is
defined and continuous on the closed interval [a, b]. Then if
γ ∈ R lies between f (a) and f (b) then there exists c ∈ [a, b]
such that γ = f (c). 1.28 To prove the Interval Value Theorem we must use the
Supremum Axiom or the Completeness Axiom.
then f is continuous on the set
x ∈ R | x(x − 1)2 6= 0 = R \ {0, 1}
1.29 The IVT allows us to say that the graph of a continuous
function f on a closed interval I is an unbroken curve on I.
= (←, 0) ∪ (0, 1) ∪ (1, →).
1.23 Theorem
Suppose that f and g : R R are such that g ◦ f : R R is
defined. Suppose also that f is continuous at a and g is (defined
and) continuous at f (a). Then g ◦ f is continuous at a. 1.24 The next two theorems are more difficult to prove.
1.25 Theorem
For all n ∈ N, the function R R : x 7→ x1/n is continuous.
1.26 Theorem
Each of the following functions is continuous:
y
y = f (x)
p
s
p
p
p
p
p
p
p
p
p
p p p p p p p p p p p p p sp
f (b)
p
p
p
p
γ s
sp
p
p
p
p
p
p
p
b
a
s
ps
ps
p
p
c
f (a) sp p p ps
x
1.30 Theorem (The Extreme Value Theorem, EVT)
Suppose that a, b ∈ R, that a < b, and that f : R R is
defined and continuous on the closed interval [a, b]. Then f has
both a minimum point and a maximum point on [a, b]
(i) exp : R → R : x 7→ ex ,
(ii) ln = loge ( = log ) : (0, →) → R,
(iii) sin : R → R,
(iv) cos : R → R.
1.8
y
y = f (x)
sp
p
p
p
p
ps
a
sp
p
ps
c
sp
p
p
p
p
p
p
p
p
p
p
p
p
p
p
sp
d
ps
p
p
p
p
p
p
p
p
p
p
p
ps
b
for all x ∈ dom( f ), if |x − a| < δ then | f (x) − f (a)| < ε.
x
1.31 We shall complete this chapter by stating Cauchy’s
(ε, δ )-definition of continuity and showing that it is equivalent
to Definition 1.8.
y
y = f (x)
sp p p p p p p p p p p p p p p p p s
ps
p
p
p
f (a) sp p p p p p p p p p p p p sp
p
p
p
ps
p
p
p
p
p
p
p
p
f (a) − ε sp p p p p p s
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
sp
sp
sp
a
I
@
@
f (a) + ε
a−δ
1.9
1.32 Definition
Suppose that f : R R and that a ∈ dom( f ). Then we say
that f satisfies Cauchy’s continuity condition at a if and only
if, for all ε > 0, there exists δ > 0, where δ depends on f , a,
and ε, such that
1.33 Less formally, we can say that f satisfies Cauchy’s
continuity condition at a if and only if by keeping x close
enough to a we can ensure that f (x) is as close as we wish to
f (a).
1.34 Theorem
Suppose that f : R R and that a ∈ dom( f ). Then f is
continuous at a if and only if f satisfies Cauchy’s continuity
condition at a.
Proof The proof is in two parts.
(i) Suppose that f satisfies Cauchy’s continuity condition at a.
We must prove that f is continuous at a.
Suppose that (an ) is a sequence in dom( f ) that converges to a.
We must prove that the sequence ( f (an )) converges to f (a).
Let ε > 0. Since f satisfies Cauchy’s continuity condition at a,
there exists δ > 0 such that, for all x ∈ dom( f ),
|x − a| < δ =⇒ | f (x) − f (a)| < ε.
x
a+δ
Since (an ) converges to a, there exists K ∈ N such that,
for all n ∈ N,
n ≥ K =⇒ |an − a| < δ .
(1)
(2)
1.10
1.11
Let n ∈ N. Then by setting δ = 1/n in (3) we see that there
exists an = x(1/n) ∈ dom( f ) such that
Since (an ) is a sequence in dom( f ), (1) and (2) imply that,
for all n ∈ N,
n ≥ K =⇒ | f (an ) − f (a)| < ε.
for all n ∈ N, |an − a| <
Therefore
(4)
and
lim( f (an )) = f (a).
for all n ∈ N, | f (an ) − f (a)| ≥ ε.
n
Therefore f is continuous at a.
(ii) We shall complete the proof of this theorem by showing
that if f does not satisfy Cauchy’s continuity condition at a
then f is not continuous at a.
Suppose that f does not satisfy Cauchy’s continuity condition
at a. This means that there exists ε > 0 such that it is not true
that there exists δ > 0 such that,
for all x ∈ dom( f ), |x − a| < δ =⇒ | f (x) − f (a)| < ε.
In other words, there exists ε > 0 such that, for all δ > 0, the
statement
for all x ∈ dom( f ), |x − a| < δ =⇒ | f (x) − f (a)| < ε.
is false.
Therefore, there exists ε > 0 such that, for all δ > 0, there
exists x(δ ) ∈ dom( f ) such that
|x(δ ) − a| < δ and | f (x(δ )) − f (a)| ≥ ε.
1
n
(3)
We write x(δ ) instead of simply x to emphasize that the value
of this number depends on the value of δ .
(5)
The sequence (1/n) converges to 0.
Therefore statement (4) and the Squeezing Rule imply that the
sequence (|an − a|) converges to 0.
Therefore (an ) converges to a.
Statement (5), on the other hand, implies that the sequence
( f (an )) does not converge to f (a). Therefore f is not
continuous at a.