Quasi-continuous function explained

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

be a topological space. A real-valued function

f:XR

is quasi-continuous at a point

x\inX

if for any

\epsilon>0

and any open neighborhood

U

of

x

there is a non-empty open set

G\subsetU

such that

|f(x)-f(y)|<\epsilon    \forally\inG

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

x\inG

.

Properties

f:XR

is continuous then

f

is quasi-continuous

f:XR

is continuous and

g:XR

is quasi-continuous, then

f+g

is quasi-continuous.

Example

Consider the function

f:RR

defined by

f(x)=0

whenever

x\leq0

and

f(x)=1

whenever

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\subsetU

such that

y<0\forally\inG

. Clearly this yields

|f(0)-f(y)|=0\forally\inG

thus f is quasi-continuous.

In contrast, the function

g:RR

defined by

g(x)=0

whenever

x

is a rational number and

g(x)=1

whenever

x

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

G

contains some

y1,y2

with

|g(y1)-g(y2)|=1

.

See also

References