In the case where and are metric spaces (e.g. Euclidean space, or the space of real numbers), a function is continuous if and only if for every and every real number , there exists a real number such that whenever a point has distance less than to , the point has distance less than to .
Continuity at a point
A related notion is that of local continuity, or continuity at a point (as opposed to the whole space at once). When and are topological spaces, we say is continuous at a point if, for every open subset containing , there is an open subset containing whose image is contained in . Of course, the function is continuous in the first sense if and only if is continuous at every point in the second sense (for students who haven’t seen this before, proving it is a worthwhile exercise).
In the common case where and are metric spaces (e.g., Euclidean spaces), a function is continuous at if and only if for every real number , there exists a real number satisfying the property that for all with . Alternatively, the function is continuous at if and only if the limit of as satisfies the equation
|Date of creation||2013-03-22 11:51:55|
|Last modified on||2013-03-22 11:51:55|
|Last modified by||djao (24)|