Recipe of Continuous Functions

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


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.


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