# Recipe of Continuous Functions

### Abstract

A function π βπ is continuous at a point π₯0,if for π>0, there exist a πΏ>0 such that |π₯βπ₯π|<πΏ implies

|π(π₯)βπ( π₯π)|<π. Therefore, the function f is a continuous function if it is continuous at every point. In the context of general topology, the βsmallβ sets may be replaced by βwell-chosen open setβ. So the definition should be thought of as: a function π is continuous at π₯ if given any small enough open set U containing π(π₯), there is an open set V (that we may be forced to choose very small) containing π₯ such that π(π)βπ. Continuity of a function depends not only upon the function π itself, but also on the topologies specified for its domain and range.

### References

2. Lipschutz S., (1965) , General Topology, Schaumβs Outline

3. Moller M. J., General Topology, Matematisk Institut, Universitetsparken 5, DK-2100 Kobenhavn

4. Munkres J., Topology, Second Edition