“Quasi-continuous function”的意思、由来-开放百科全书

词条

Quasi-continuous function

释义

Definition
Properties
Example
References

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 be a topological space. A real-valued function is quasi-continuous at a point if for any every and any open neighborhood of there is a non-empty open set such that

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

Properties

If is continuous then is quasi-continuous
If is continuous and is quasi-continuous, then is quasi-continuous.

Example

Consider the function defined by whenever and whenever . 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 such that . Clearly this yields thus f is quasi-continuous.

References

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2 : Calculus|Continuous mappings

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