continuous (original) (raw)

In the case where X and Y are metric spaces (e.g. Euclidean space, or the space of real numbers), a function f:X→Y is continuous if and only if for every x∈X and every real number ϵ>0, there exists a real number δ>0 such that whenever a point z∈X has distance less than δ to x, the point f⁢(z)∈Y has distance less than ϵ to f⁢(x).

A related notion is that of local continuity, or continuity at a point (as opposed to the whole space X at once). When X and Y are topological spaces, we say f is continuous at a point x∈X if, for every open subset V⊂Y containing f⁢(x), there is an open subset U⊂X containing x whose image f⁢(U) is contained in V. Of course, the function f:X→Y is continuous in the first sense if and only if f is continuous at every point x∈X in the second sense (for students who haven’t seen this before, proving it is a worthwhile exercise).

In the common case where X and Y are metric spaces (e.g., Euclidean spaces), a function f is continuous at x∈X if and only if for every real number ϵ>0, there exists a real number δ>0 satisfying the property that dY⁢(f⁢(x),f⁢(z))<ϵ for all z∈X with dX⁢(x,z)<δ. Alternatively, the function f is continuous at a∈X if and only if the limit of f⁢(x) as x→a satisfies the equation