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.
Let
X
be a topological space. A real-valued function
f
:
X
→
R
is quasi-continuous at a point
x
∈
X
if for any every
ϵ
>
0
and any open neighborhood
U
of
x
there is a non-empty open set
G
⊂
U
such that
|
f
(
x
)
−
f
(
y
)
|
<
ϵ
∀
y
∈
G
Note that in the above definition, it is not necessary that
x
∈
G
.
If
f
:
X
→
R
is continuous then
f
is quasi-continuous
If
f
:
X
→
R
is continuous and
g
:
X
→
R
is quasi-continuous, then
f
+
g
is quasi-continuous.
Consider the function
f
:
R
→
R
defined by
f
(
x
)
=
0
whenever
x
≤
0
and
f
(
x
)
=
1
whenever
x
>
0
. Clearly f is continuous everywhere except at x=0, thus quasi-continuous everywhere except at x=0. At x=0, take any open neighborhood U of x. Then there exists an open set
G
⊂
U
such that
y
<
0
∀
y
∈
G
. Clearly this yields
|
f
(
0
)
−
f
(
y
)
|
=
0
∀
y
∈
G
thus f is quasi-continuous.
