• Source: Quasi-continuous function

In mathematics, the notion of a quasi-continuous function is similar to, but weaker than, the notion of a continuous function. All continuous functions are quasi-continuous but the converse is not true in general.




Definition


Let



X


{\displaystyle X}

be a topological space. A real-valued function



f
:
X
→

R



{\displaystyle f:X\rightarrow \mathbb {R} }

is quasi-continuous at a point



x
∈
X


{\displaystyle x\in X}

if for any



ϵ
>
0


{\displaystyle \epsilon >0}

and any open neighborhood



U


{\displaystyle U}

of



x


{\displaystyle x}

there is a non-empty open set



G
⊂
U


{\displaystyle G\subset U}

such that





|

f
(
x
)
−
f
(
y
)

|

<
ϵ




∀
y
∈
G


{\displaystyle |f(x)-f(y)|<\epsilon \;\;\;\;\forall y\in G}


Note that in the above definition, it is not necessary that



x
∈
G


{\displaystyle x\in G}

.




Properties


If



f
:
X
→

R



{\displaystyle f:X\rightarrow \mathbb {R} }

is continuous then



f


{\displaystyle f}

is quasi-continuous
If



f
:
X
→

R



{\displaystyle f:X\rightarrow \mathbb {R} }

is continuous and



g
:
X
→

R



{\displaystyle g:X\rightarrow \mathbb {R} }

is quasi-continuous, then



f
+
g


{\displaystyle f+g}

is quasi-continuous.




Example


Consider the function



f
:

R

→

R



{\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }

defined by



f
(
x
)
=
0


{\displaystyle f(x)=0}

whenever



x
≤
0


{\displaystyle x\leq 0}

and



f
(
x
)
=
1


{\displaystyle f(x)=1}

whenever



x
>
0


{\displaystyle x>0}

. Clearly f is continuous everywhere except at x=0, thus quasi-continuous everywhere except (at most) at x=0. At x=0, take any open neighborhood U of x. Then there exists an open set



G
⊂
U


{\displaystyle G\subset U}

such that



y
<
0

∀
y
∈
G


{\displaystyle y<0\;\forall y\in G}

. Clearly this yields




|

f
(
0
)
−
f
(
y
)

|

=
0

∀
y
∈
G


{\displaystyle |f(0)-f(y)|=0\;\forall y\in G}

thus f is quasi-continuous.
In contrast, the function



g
:

R

→

R



{\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} }

defined by



g
(
x
)
=
0


{\displaystyle g(x)=0}

whenever



x


{\displaystyle x}

is a rational number and



g
(
x
)
=
1


{\displaystyle g(x)=1}

whenever



x


{\displaystyle x}

is an irrational number is nowhere quasi-continuous, since every nonempty open set



G


{\displaystyle G}

contains some




y

1


,

y

2




{\displaystyle y_{1},y_{2}}

with




|

g
(

y

1


)
−
g
(

y

2


)

|

=
1


{\displaystyle |g(y_{1})-g(y_{2})|=1}

.




See also


Cliquish function




References


Ján Borsík (2007–2008). "Points of Continuity, Quasi-continuity, cliquishness, and Upper and Lower Quasi-continuity". Real Analysis Exchange. 33 (2): 339–350.
T. Neubrunn (1988). "Quasi-continuity". Real Analysis Exchange. 14 (2): 259–308. JSTOR 44151947.

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