• Source: Locally constant function

In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function.




Definition


Let



f
:
X
→
S


{\displaystyle f:X\to S}

be a function from a topological space



X


{\displaystyle X}

into a set



S
.


{\displaystyle S.}


If



x
∈
X


{\displaystyle x\in X}

then



f


{\displaystyle f}

is said to be locally constant at



x


{\displaystyle x}

if there exists a neighborhood



U
⊆
X


{\displaystyle U\subseteq X}

of



x


{\displaystyle x}

such that



f


{\displaystyle f}

is constant on



U
,


{\displaystyle U,}

which by definition means that



f
(
u
)
=
f
(
v
)


{\displaystyle f(u)=f(v)}

for all



u
,
v
∈
U
.


{\displaystyle u,v\in U.}


The function



f
:
X
→
S


{\displaystyle f:X\to S}

is called locally constant if it is locally constant at every point



x
∈
X


{\displaystyle x\in X}

in its domain.




Examples


Every constant function is locally constant. The converse will hold if its domain is a connected space.
Every locally constant function from the real numbers




R



{\displaystyle \mathbb {R} }

to




R



{\displaystyle \mathbb {R} }

is constant, by the connectedness of




R

.


{\displaystyle \mathbb {R} .}

But the function



f
:

Q

→

R



{\displaystyle f:\mathbb {Q} \to \mathbb {R} }

from the rationals




Q



{\displaystyle \mathbb {Q} }

to




R

,


{\displaystyle \mathbb {R} ,}

defined by



f
(
x
)
=
0

for

x
<
π
,


{\displaystyle f(x)=0{\text{ for }}x<\pi ,}

and



f
(
x
)
=
1

for

x
>
π
,


{\displaystyle f(x)=1{\text{ for }}x>\pi ,}

is locally constant (this uses the fact that



π


{\displaystyle \pi }

is irrational and that therefore the two sets



{
x
∈

Q

:
x
<
π
}


{\displaystyle \{x\in \mathbb {Q} :x<\pi \}}

and



{
x
∈

Q

:
x
>
π
}


{\displaystyle \{x\in \mathbb {Q} :x>\pi \}}

are both open in




Q



{\displaystyle \mathbb {Q} }

).
If



f
:
A
→
B


{\displaystyle f:A\to B}

is locally constant, then it is constant on any connected component of



A
.


{\displaystyle A.}

The converse is true for locally connected spaces, which are spaces whose connected components are open subsets.
Further examples include the following:

Given a covering map



p
:
C
→
X
,


{\displaystyle p:C\to X,}

then to each point



x
∈
X


{\displaystyle x\in X}

we can assign the cardinality of the fiber




p

−
1


(
x
)


{\displaystyle p^{-1}(x)}

over



x


{\displaystyle x}

; this assignment is locally constant.
A map from a topological space



A


{\displaystyle A}

to a discrete space



B


{\displaystyle B}

is continuous if and only if it is locally constant.




Connection with sheaf theory


There are sheaves of locally constant functions on



X
.


{\displaystyle X.}

To be more definite, the locally constant integer-valued functions on



X


{\displaystyle X}

form a sheaf in the sense that for each open set



U


{\displaystyle U}

of



X


{\displaystyle X}

we can form the functions of this kind; and then verify that the sheaf axioms hold for this construction, giving us a sheaf of abelian groups (even commutative rings). This sheaf could be written




Z

X




{\displaystyle Z_{X}}

; described by means of stalks we have stalk




Z

x


,


{\displaystyle Z_{x},}

a copy of



Z


{\displaystyle Z}

at



x
,


{\displaystyle x,}

for each



x
∈
X
.


{\displaystyle x\in X.}

This can be referred to a constant sheaf, meaning exactly sheaf of locally constant functions taking their values in the (same) group. The typical sheaf of course is not constant in this way; but the construction is useful in linking up sheaf cohomology with homology theory, and in logical applications of sheaves. The idea of local coefficient system is that we can have a theory of sheaves that locally look like such 'harmless' sheaves (near any



x


{\displaystyle x}

), but from a global point of view exhibit some 'twisting'.




See also


Liouville's theorem (complex analysis) – Theorem in complex analysis
Locally constant sheaf




References

Kata Kunci Pencarian:

• Locally constant function
• Lipschitz continuity
• Constant function
• Local boundedness
• Locally constant sheaf
• Step function
• Locally integrable function
• Constant sheaf
• Constant of integration
• Barsotti–Tate group