- Source: Nowhere continuous function
In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If
f
{\displaystyle f}
is a function from real numbers to real numbers, then
f
{\displaystyle f}
is nowhere continuous if for each point
x
{\displaystyle x}
there is some
ε
>
0
{\displaystyle \varepsilon >0}
such that for every
δ
>
0
,
{\displaystyle \delta >0,}
we can find a point
y
{\displaystyle y}
such that
|
x
−
y
|
<
δ
{\displaystyle |x-y|<\delta }
and
|
f
(
x
)
−
f
(
y
)
|
≥
ε
{\displaystyle |f(x)-f(y)|\geq \varepsilon }
. Therefore, no matter how close it gets to any fixed point, there are even closer points at which the function takes not-nearby values.
More general definitions of this kind of function can be obtained, by replacing the absolute value by the distance function in a metric space, or by using the definition of continuity in a topological space.
Examples
= Dirichlet function
=One example of such a function is the indicator function of the rational numbers, also known as the Dirichlet function. This function is denoted as
1
Q
{\displaystyle \mathbf {1} _{\mathbb {Q} }}
and has domain and codomain both equal to the real numbers. By definition,
1
Q
(
x
)
{\displaystyle \mathbf {1} _{\mathbb {Q} }(x)}
is equal to
1
{\displaystyle 1}
if
x
{\displaystyle x}
is a rational number and it is
0
{\displaystyle 0}
otherwise.
More generally, if
E
{\displaystyle E}
is any subset of a topological space
X
{\displaystyle X}
such that both
E
{\displaystyle E}
and the complement of
E
{\displaystyle E}
are dense in
X
,
{\displaystyle X,}
then the real-valued function which takes the value
1
{\displaystyle 1}
on
E
{\displaystyle E}
and
0
{\displaystyle 0}
on the complement of
E
{\displaystyle E}
will be nowhere continuous. Functions of this type were originally investigated by Peter Gustav Lejeune Dirichlet.
= Non-trivial additive functions
=A function
f
:
R
→
R
{\displaystyle f:\mathbb {R} \to \mathbb {R} }
is called an additive function if it satisfies Cauchy's functional equation:
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
for all
x
,
y
∈
R
.
{\displaystyle f(x+y)=f(x)+f(y)\quad {\text{ for all }}x,y\in \mathbb {R} .}
For example, every map of form
x
↦
c
x
,
{\displaystyle x\mapsto cx,}
where
c
∈
R
{\displaystyle c\in \mathbb {R} }
is some constant, is additive (in fact, it is linear and continuous). Furthermore, every linear map
L
:
R
→
R
{\displaystyle L:\mathbb {R} \to \mathbb {R} }
is of this form (by taking
c
:=
L
(
1
)
{\displaystyle c:=L(1)}
).
Although every linear map is additive, not all additive maps are linear. An additive map
f
:
R
→
R
{\displaystyle f:\mathbb {R} \to \mathbb {R} }
is linear if and only if there exists a point at which it is continuous, in which case it is continuous everywhere. Consequently, every non-linear additive function
R
→
R
{\displaystyle \mathbb {R} \to \mathbb {R} }
is discontinuous at every point of its domain.
Nevertheless, the restriction of any additive function
f
:
R
→
R
{\displaystyle f:\mathbb {R} \to \mathbb {R} }
to any real scalar multiple of the rational numbers
Q
{\displaystyle \mathbb {Q} }
is continuous; explicitly, this means that for every real
r
∈
R
,
{\displaystyle r\in \mathbb {R} ,}
the restriction
f
|
r
Q
:
r
Q
→
R
{\displaystyle f{\big \vert }_{r\mathbb {Q} }:r\,\mathbb {Q} \to \mathbb {R} }
to the set
r
Q
:=
{
r
q
:
q
∈
Q
}
{\displaystyle r\,\mathbb {Q} :=\{rq:q\in \mathbb {Q} \}}
is a continuous function.
Thus if
f
:
R
→
R
{\displaystyle f:\mathbb {R} \to \mathbb {R} }
is a non-linear additive function then for every point
x
∈
R
,
{\displaystyle x\in \mathbb {R} ,}
f
{\displaystyle f}
is discontinuous at
x
{\displaystyle x}
but
x
{\displaystyle x}
is also contained in some dense subset
D
⊆
R
{\displaystyle D\subseteq \mathbb {R} }
on which
f
{\displaystyle f}
's restriction
f
|
D
:
D
→
R
{\displaystyle f\vert _{D}:D\to \mathbb {R} }
is continuous (specifically, take
D
:=
x
Q
{\displaystyle D:=x\,\mathbb {Q} }
if
x
≠
0
,
{\displaystyle x\neq 0,}
and take
D
:=
Q
{\displaystyle D:=\mathbb {Q} }
if
x
=
0
{\displaystyle x=0}
).
= Discontinuous linear maps
=A linear map between two topological vector spaces, such as normed spaces for example, is continuous (everywhere) if and only if there exists a point at which it is continuous, in which case it is even uniformly continuous. Consequently, every linear map is either continuous everywhere or else continuous nowhere.
Every linear functional is a linear map and on every infinite-dimensional normed space, there exists some discontinuous linear functional.
= Other functions
=The Conway base 13 function is discontinuous at every point.
Hyperreal characterisation
A real function
f
{\displaystyle f}
is nowhere continuous if its natural hyperreal extension has the property that every
x
{\displaystyle x}
is infinitely close to a
y
{\displaystyle y}
such that the difference
f
(
x
)
−
f
(
y
)
{\displaystyle f(x)-f(y)}
is appreciable (that is, not infinitesimal).
See also
Blumberg theorem – even if a real function
f
:
R
→
R
{\displaystyle f:\mathbb {R} \to \mathbb {R} }
is nowhere continuous, there is a dense subset
D
{\displaystyle D}
of
R
{\displaystyle \mathbb {R} }
such that the restriction of
f
{\displaystyle f}
to
D
{\displaystyle D}
is continuous.
Thomae's function (also known as the popcorn function) – a function that is continuous at all irrational numbers and discontinuous at all rational numbers.
Weierstrass function – a function continuous everywhere (inside its domain) and differentiable nowhere.
References
External links
"Dirichlet-function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Dirichlet Function — from MathWorld
The Modified Dirichlet Function Archived 2019-05-02 at the Wayback Machine by George Beck, The Wolfram Demonstrations Project.
Kata Kunci Pencarian:
- Nowhere continuous function
- Weierstrass function
- Differentiable function
- Continuous function
- Quasi-continuous function
- Baire function
- List of types of functions
- Absolute continuity
- Pathological (mathematics)
- Uniform continuity
