Recipe of Continuous Functions

  • Abraham Quansah
  • Nicholas Adjei-Boateng
  • Charlotte Anima Nketia KNUST
Keywords: Continuous function, Mapping, Topology, Neighborhood, Domain, Co-domain

Abstract

A function 𝑅→𝑅 is continuous at a point π‘₯0,if for πœ€>0, there exist a 𝛿>0 such that |π‘₯βˆ’π‘₯π‘œ|<𝛿 implies

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

References

Adams C. and Franzosa R.,(2008) , Introduction To Topology, Pure an Applied
2. Lipschutz S., (1965) , General Topology, Schaum’s Outline
3. Moller M. J., General Topology, Matematisk Institut, Universitetsparken 5, DK-2100 Kobenhavn
4. Munkres J., Topology, Second Edition
Published
2017-09-03