Real-valued continuous functions Continuous function

1 real-valued continuous functions

1.1 definition

1.1.1 definition in terms of limits of functions
1.1.2 definition in terms of neighborhoods
1.1.3 definition in terms of limits of sequences
1.1.4 weierstrass , jordan definitions (epsilon–delta) of continuous functions
1.1.5 definition in terms of control of remainder
1.1.6 definition using oscillation
1.1.7 definition using hyperreals


1.2 examples
1.3 examples of discontinuous functions
1.4 properties

1.4.1 intermediate value theorem
1.4.2 extreme value theorem
1.4.3 relation differentiability , integrability
1.4.4 pointwise , uniform limits


1.5 directional , semi-continuity

real-valued continuous functions
definition

the function



f
(
x
)
=



1
x





{\displaystyle f(x)={\tfrac {1}{x}}}

continuous on whole domain




r

∖
{
0
}


{\displaystyle \mathbb {r} \setminus \{0\}}

, not continuous on domain




r



{\displaystyle \mathbb {r} }

because discontinuous @



x
=
0


{\displaystyle x=0}

a function set of real numbers real numbers can represented graph in cartesian plane; such function continuous if, speaking, graph single unbroken curve domain entire real line. more mathematically rigorous definition given below.

the rigorous definition of real valued function of real variable given in first course in calculus in terms of idea of limit. first, function said continuous @ point on real line if limit x approaches point equal function value @ point. function said continuous if continuous @ every point. point function not continuous called discontinuity.

there several different definitions of continuous function. function said continuous if continuous @ every point in domain. in case, function y=tan(x) continuous. exception made endpoints, that, example if graph has left-hand endpoint, limit right required equal value of function. in case y=√x continuous. common , restrictive definition function continuous if continuous @ real numbers. in case, previous 2 examples not continuous, every polynomial function continuous, sine, cosine, , exponential functions. care should exercised in using word continuous, clear meaning of word intended.

using mathematical notation, there several ways define continuous functions in each of 3 senses mentioned above.

let

f
:
d
→

r

.


{\displaystyle f\colon d\rightarrow \mathbf {r} .}

function defined on subset d of set r of real numbers. subset d domain of f. possible choices include d=r, whole set of real numbers, d open interval




d
=
(
a
,
b
)
=
{
x
∈

r



|


a
<
x
<
b
}
,


{\displaystyle d=(a,b)=\{x\in \mathbf {r} \,|\,a<x<b\},}

, or d closed interval




d
=
[
a
,
b
]
=
{
x
∈

r



|


a
≤
x
≤
b
}
.


{\displaystyle d=[a,b]=\{x\in \mathbf {r} \,|\,a\leq x\leq b\}.}

here, , b real numbers.

definition in terms of limits of functions

the function f continuous @ point c of domain if limit of f(x) x approaches c through domain of f exists , equal f(c). in mathematical notation, written as

lim

x
→
c



f
(
x
)

=
f
(
c
)
.


{\displaystyle \lim _{x\to c}{f(x)}=f(c).}

in detail means 3 conditions: first, f has defined @ c. second, limit on left hand side of equation has exist. third, value of limit must equal f(c).

(we have here assumed domain of f not have isolated points. example, interval or union of intervals has no isolated points.)

definition in terms of neighborhoods

a neighborhood of point c set contains points of domain within fixed distance of c. intuitively, function continuous @ point c if range of restriction of f neighborhood of c shrinks single point f(c) width of neighborhood around c shrinks zero. more precisely, function f continuous @ point c of domain if, neighborhood




n

1


(
f
(
c
)
)


{\displaystyle n_{1}(f(c))}

there neighborhood




n

2


(
c
)


{\displaystyle n_{2}(c)}

such



f
(
x
)
∈

n

1


(
f
(
c
)
)


{\displaystyle f(x)\in n_{1}(f(c))}

whenever



x
∈

n

2


(
c
)


{\displaystyle x\in n_{2}(c)}

.

this definition requires domain , codomain topological spaces , general definition. follows definition function f automatically continuous @ every isolated point of domain. specific example, every real valued function on set of integers continuous.

definition in terms of limits of sequences

the sequence exp(1/n) converges exp(0)

one can instead require sequence



(

x

n



)

n
∈

n





{\displaystyle (x_{n})_{n\in \mathbb {n} }}

of points in domain converges c, corresponding sequence





(
f
(

x

n


)
)


n
∈

n





{\displaystyle \left(f(x_{n})\right)_{n\in \mathbb {n} }}

converges f(c). in mathematical notation,



∀
(

x

n



)

n
∈

n



⊂
d
:

lim

n
→
∞



x

n


=
c
⇒

lim

n
→
∞


f
(

x

n


)
=
f
(
c
)

.


{\displaystyle \forall (x_{n})_{n\in \mathbb {n} }\subset d:\lim _{n\to \infty }x_{n}=c\rightarrow \lim _{n\to \infty }f(x_{n})=f(c)\,.}

weierstrass , jordan definitions (epsilon–delta) of continuous functions

illustration of ε-δ-definition: ε=0.5, c=2, value δ=0.5 satisfies condition of definition.

explicitly including definition of limit of function, obtain self-contained definition: given function f : d → r above , element x0 of domain d, f said continuous @ point x0 when following holds: number ε > 0, small, there exists number δ > 0 such x in domain of f x0 − δ < x < x0 + δ, value of f(x) satisfies

f
(

x

0


)
−
ε
<
f
(
x
)
<
f
(

x

0


)
+
ε
.


{\displaystyle f(x_{0})-\varepsilon <f(x)<f(x_{0})+\varepsilon .}

alternatively written, continuity of f : d → r @ x0 ∈ d means every ε > 0 there exists δ > 0 such x ∈ d :

|

x
−

x

0



|

<
δ
⇒

|

f
(
x
)
−
f
(

x

0


)

|

<
ε
.


{\displaystyle |x-x_{0}|<\delta \rightarrow |f(x)-f(x_{0})|<\varepsilon .}

more intuitively, can if want f(x) values stay in small neighborhood around f(x0), need choose small enough neighborhood x values around x0. if can no matter how small f(x) neighborhood is, f continuous at x0.

in modern terms, generalized definition of continuity of function respect basis topology, here metric topology.

weierstrass had required interval x0 − δ < x < x0 + δ entirely within domain d, jordan removed restriction.

definition in terms of control of remainder

in proofs , numerical analysis need know how fast limits converging, or in other words, control of remainder. can formalise definition of continuity. function



c
:
[
0
,
∞
)
→
[
0
,
∞
]


{\displaystyle c:[0,\infty )\to [0,\infty ]}

called control function if

c non decreasing





inf

δ
>
0


c
(
δ
)
=
0


{\displaystyle \inf _{\delta >0}c(\delta )=0}

a function f : d → r c-continuous @ x0 if

|

f
(
x
)
−
f
(

x

0


)

|

≤
c
(

|

x
−

x

0



|

)


{\displaystyle |f(x)-f(x_{0})|\leq c(|x-x_{0}|)}





x
∈
d


{\displaystyle x\in d}

a function continuous in x0 if c-continuous control function c.

this approach leads naturally refining notion of continuity restricting set of admissible control functions. given set of control functions





c




{\displaystyle {\mathcal {c}}}

function





c




{\displaystyle {\mathcal {c}}}

continuous if c continuous



c
∈


c




{\displaystyle c\in {\mathcal {c}}}

. example lipschitz , hölder continuous functions of exponent α below defined set of control functions

c




l
i
p
s
c
h
i
t
z



=
{
c

|

c
(
δ
)
=
k

|

δ

|

,
 
k
>
0
}


{\displaystyle {\mathcal {c}}_{\mathrm {lipschitz} }=\{c|c(\delta )=k|\delta |,\ k>0\}}

respectively

c




h
o
l
d
e
r

−
α


=


{\displaystyle {\mathcal {c}}_{\mathrm {holder} -\alpha }=}





{
c

|

c
(
δ
)
=
k

|

δ


|


α


,
 
k
>
0
}


{\displaystyle \{c|c(\delta )=k|\delta |^{\alpha },\ k>0\}}

.



definition using oscillation

the failure of function continuous @ point quantified oscillation.

continuity can defined in terms of oscillation: function f continuous @ point x0 if , if oscillation @ point zero; in symbols,




ω

f


(

x

0


)
=
0.


{\displaystyle \omega _{f}(x_{0})=0.}

benefit of definition quantifies discontinuity: oscillation gives how function discontinuous @ point.

this definition useful in descriptive set theory study set of discontinuities , continuous points – continuous points intersection of sets oscillation less ε (hence gδ set) – , gives quick proof of 1 direction of lebesgue integrability condition.

the oscillation equivalent ε-δ definition simple re-arrangement, , using limit (lim sup, lim inf) define oscillation: if (at given point) given ε0 there no δ satisfies ε-δ definition, oscillation @ least ε0, , conversely if every ε there desired δ, oscillation 0. oscillation definition can naturally generalized maps topological space metric space.

definition using hyperreals

cauchy defined continuity of function in following intuitive terms: infinitesimal change in independent variable corresponds infinitesimal change of dependent variable (see cours d analyse, page 34). non-standard analysis way of making mathematically rigorous. real line augmented addition of infinite , infinitesimal numbers form hyperreal numbers. in nonstandard analysis, continuity can defined follows.

a real-valued function f continuous @ x if natural extension hyperreals has property infinitesimal dx, f(x+dx) − f(x) infinitesimal

(see microcontinuity). in other words, infinitesimal increment of independent variable produces infinitesimal change of dependent variable, giving modern expression augustin-louis cauchy s definition of continuity.

examples

the graph of cubic function has no jumps or holes. function continuous.

all polynomial functions, such f(x) = x + x - 5x + 3 (pictured), continuous. consequence of fact that, given 2 continuous functions

f
,
g
:
d
→

r



{\displaystyle f,g\colon d\rightarrow \mathbf {r} }

defined on same domain d, sum f + g , product fg of 2 functions continuous (on same domain d). moreover, function

f
g


:
{
x
∈
d

|

g
(
x
)
≠
0
}
→

r

,
x
↦



f
(
x
)


g
(
x
)





{\displaystyle {\frac {f}{g}}\colon \{x\in d|g(x)\neq 0\}\rightarrow \mathbf {r} ,x\mapsto {\frac {f(x)}{g(x)}}}

is continuous. (the points g(x) 0 discarded, not in domain of f/g.) example, function (pictured)

f
(
x
)
=



2
x
−
1


x
+
2





{\displaystyle f(x)={\frac {2x-1}{x+2}}}




the graph of continuous rational function. function not defined x=−2. vertical , horizontal lines asymptotes.

is defined real numbers x ≠ −2 , continuous @ every such point. continuous function. question of continuity @ x = −2 not arise, since x = −2 not in domain of f. there no continuous function f: r → r agrees f(x) x ≠ −2. sinc function g(x) = (sin x)/x, defined x≠0 continuous @ these points. continuous function, too. however, unlike 1 of previous example, 1 can extended continuous function on real numbers, namely

g
(
x
)
=


{






sin
⁡
(
x
)

x





 if 

x
≠
0




1



 if 

x
=
0
,








{\displaystyle g(x)={\begin{cases}{\frac {\sin(x)}{x}}&{\text{ if }}x\neq 0\\1&{\text{ if }}x=0,\end{cases}}}

since limit of g(x), when x approaches 0, 1. therefore, point x=0 called removable singularity of g.

given 2 continuous functions

f
:
i
→
j
(
⊂

r

)
,
g
:
j
→

r

,


{\displaystyle f\colon i\rightarrow j(\subset \mathbf {r} ),g\colon j\rightarrow \mathbf {r} ,}

the composition

g
∘
f
:
i
→

r

,
x
↦
g
(
f
(
x
)
)


{\displaystyle g\circ f\colon i\rightarrow \mathbf {r} ,x\mapsto g(f(x))}

is continuous.

examples of discontinuous functions

plot of signum function. shows




lim

n
→
∞


sgn
⁡

(



1
n



)

≠
sgn
⁡

(

lim

n
→
∞





1
n



)



{\displaystyle \lim _{n\to \infty }\operatorname {sgn} \left({\tfrac {1}{n}}\right)\neq \operatorname {sgn} \left(\lim _{n\to \infty }{\tfrac {1}{n}}\right)}

. thus, signum function not continuous @ point 0.

an example of discontinuous function heaviside step function



h


{\displaystyle h}

, defined by

h
(
x
)
=


{



1



 if 

x
≥
0




0



 if 

x
<
0








{\displaystyle h(x)={\begin{cases}1&{\text{ if }}x\geq 0\\0&{\text{ if }}x<0\end{cases}}}

pick instance



ϵ
=


1
2




{\displaystyle \epsilon ={\frac {1}{2}}}

. there no



δ


{\displaystyle \delta }

-neighborhood around



x
=
0


{\displaystyle x=0}

force



h
(
x
)


{\displaystyle h(x)}

values within



ϵ


{\displaystyle \epsilon }

of



h
(
0
)


{\displaystyle h(0)}

. intuitively can think of type of discontinuity sudden jump in function values.

similarly, signum or sign function

sgn
⁡
(
x
)
=


{



1



 if 

x
>
0




0



 if 

x
=
0




−
1



 if 

x
<
0








{\displaystyle \operatorname {sgn}(x)={\begin{cases}1&{\text{ if }}x>0\\0&{\text{ if }}x=0\\-1&{\text{ if }}x<0\end{cases}}}

is discontinuous @



x
=
0


{\displaystyle x=0}

continuous everywhere else. yet example: function

f
(
x
)
=


{



sin
⁡

(


1

x

2




)


 if 

x
≠
0




0

 if 

x
=
0








{\displaystyle f(x)={\begin{cases}\sin \left({\frac {1}{x^{2}}}\right){\text{ if }}x\neq 0\\0{\text{ if }}x=0\end{cases}}}

is continuous everywhere apart



x
=
0


{\displaystyle x=0}

.

plot of thomae s function domain



0
<
x
<
1


{\displaystyle 0<x<1}

.

thomae s function,

f
(
x
)
=


{



1

 if 

x
=
0






1
q



 if 

x
=


p
q



(in lowest terms) rational number





0

 if 

x

 is irrational

.








{\displaystyle f(x)={\begin{cases}1{\text{ if }}x=0\\{\frac {1}{q}}{\text{ if }}x={\frac {p}{q}}{\text{(in lowest terms) rational number}}\\0{\text{ if }}x{\text{ irrational}}.\end{cases}}}

is continuous @ irrational numbers , discontinuous @ rational numbers. in similar vein, dirichlet s function

d
(
x
)
=


{



0

 if 

x

 is irrational 

(
∈

r

∖

q

)




1

 if 

x

 is rational 

(
∈

q

)








{\displaystyle d(x)={\begin{cases}0{\text{ if }}x{\text{ irrational }}(\in \mathbb {r} \setminus \mathbb {q} )\\1{\text{ if }}x{\text{ rational }}(\in \mathbb {q} )\end{cases}}}

is continuous.

properties
intermediate value theorem

the intermediate value theorem existence theorem, based on real number property of completeness, , states:

if real-valued function f continuous on closed interval [a, b] , k number between f(a) , f(b), there number c in [a, b] such f(c) = k.

for example, if child grows 1 m 1.5 m between ages of 2 , 6 years, then, @ time between 2 , 6 years of age, child s height must have been 1.25 m.

as consequence, if f continuous on [a, b] , f(a) , f(b) differ in sign, then, @ point c in [a, b], f(c) must equal zero.

extreme value theorem

the extreme value theorem states if function f defined on closed interval [a,b] (or closed , bounded set) , continuous there, function attains maximum, i.e. there exists c ∈ [a,b] f(c) ≥ f(x) x ∈ [a,b]. same true of minimum of f. these statements not, in general, true if function defined on open interval (a,b) (or set not both closed , bounded), as, example, continuous function f(x) = 1/x, defined on open interval (0,1), not attain maximum, being unbounded above.

relation differentiability , integrability

every differentiable function

f
:
(
a
,
b
)
→

r



{\displaystyle f\colon (a,b)\rightarrow \mathbf {r} }

is continuous, can shown. converse not hold: example, absolute value function

f
(
x
)
=

|

x

|

=


{



x

 if 

x
≥
0




−
x

 if 

x
<
0








{\displaystyle f(x)=|x|={\begin{cases}x{\text{ if }}x\geq 0\\-x{\text{ if }}x<0\end{cases}}}

is everywhere continuous. however, not differentiable @ x = 0 (but everywhere else). weierstrass s function everywhere continuous differentiable.

the derivative f′(x) of differentiable function f(x) need not continuous. if f′(x) continuous, f(x) said continuously differentiable. set of such functions denoted c((a, b)). more generally, set of functions

f
:
Ω
→

r



{\displaystyle f\colon \omega \rightarrow \mathbf {r} }

(from open interval (or open subset of r) Ω reals) such f n times differentiable , such n-th derivative of f continuous denoted c(Ω). see differentiability class. in field of computer graphics, these 3 levels called g (continuity of position), g (continuity of tangency), , g (continuity of curvature).

every continuous function

f
:
[
a
,
b
]
→

r



{\displaystyle f\colon [a,b]\rightarrow \mathbf {r} }

is integrable (for example in sense of riemann integral). converse not hold, (integrable, discontinuous) sign function shows.

pointwise , uniform limits

a sequence of continuous functions fn(x) (pointwise) limit function f(x) discontinuous. convergence not uniform.

given sequence

f

1


,

f

2


,
…
:
i
→

r



{\displaystyle f_{1},f_{2},\dotsc \colon i\rightarrow \mathbf {r} }

of functions such limit

f
(
x
)
:=

lim

n
→
∞



f

n


(
x
)


{\displaystyle f(x):=\lim _{n\rightarrow \infty }f_{n}(x)}

exists x in d, resulting function f(x) referred pointwise limit of sequence of functions (fn)n∈n. pointwise limit function need not continuous, if functions fn continuous, animation @ right shows. however, f continuous when sequence converges uniformly, uniform convergence theorem. theorem can used show exponential functions, logarithms, square root function, trigonometric functions continuous.

directional , semi-continuity

discontinuous functions may discontinuous in restricted way, giving rise concept of directional continuity (or right , left continuous functions) , semi-continuity. speaking, function right-continuous if no jump occurs when limit point approached right. formally, f said right-continuous @ point c if following holds: number ε > 0 small, there exists number δ > 0 such x in domain c < x < c + δ, value of f(x) satisfy

|

f
(
x
)
−
f
(
c
)

|

<
ε
.


{\displaystyle |f(x)-f(c)|<\varepsilon .}

this same condition continuous functions, except required hold x strictly larger c only. requiring instead x c − δ < x < c yields notion of left-continuous functions. function continuous if , if both right-continuous , left-continuous.

a function f lower semi-continuous if, roughly, jumps might occur go down, not up. is, ε > 0, there exists number δ > 0 such x in domain |x − c| < δ, value of f(x) satisfies

f
(
x
)
≥
f
(
c
)
−
ϵ
.


{\displaystyle f(x)\geq f(c)-\epsilon .}

the reverse condition upper semi-continuity.