Content deleted Content added
No edit summary |
|||
Line 24:
In general, if ''E'' is any subset of a [[topological space]] ''X'' such that both ''E'' and the complement of ''E'' are dense in ''X'', then the real-valued function which takes the value 1 on ''E'' and 0 on the complement of ''E'' will be nowhere continuous. Functions of this type were originally investigated by [[Johann Peter Gustav Lejeune Dirichlet|Dirichlet]].
==Hyperreal characterisation==
A real function ''f'' is nowhere continuous if its natural hyperreal extension has the property that every ''x'' is infinitely close to a ''y'' such that the difference ''f(x)-f(y)'' is appreciable (i.e., not [[infinitesimal]]).
==See also==
| |||