# Defining continuity of real functions of real variables

@article{Harper2016DefiningCO, title={Defining continuity of real functions of real variables}, author={John F. Harper}, journal={BSHM Bulletin: Journal of the British Society for the History of Mathematics}, year={2016}, volume={31}, pages={189 - 204} }

Continuity of a real function of a real variable has been defined in various ways over almost 200 years. Contrary to popular belief, the definitions are not all equivalent, because their consequences for four somewhat pathological functions reveal five essentially different cases. The four defensible ones imply just two cases for continuity on an interval if that is defined by using pointwise continuity at each point. Some authors had trouble: two different textbooks each gave two arguably…

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