In mathematics, the Heine–Cantor theorem states that a continuous function between two metric spaces is uniformly continuous if its domain is compact.
The theorem is named after Eduard Heine and Georg Cantor.
is a continuous function between two metric spaces
is uniformly continuous.
An important special case of the Cantor theorem is that every continuous function from a closed bounded interval to the real numbers is uniformly continuous.
are two metric spaces with metrics
Suppose further that a function
is uniformly continuous, that is, for every positive real number
there exists a positive real number
δ > 0
in the function domain
( x , y ) < δ
Consider some positive real number
By continuity, for any point
, there exists some positive real number
δ
δ
, i.e. the set Since each point
, we find that the collection
is an open cover of
is compact, this cover has a finite subcover
Each of these open sets has an associated radius
, i.e. the minimum radius of these open sets.
Since we have a finite number of positive radii, this minimum
is well-defined and positive.
works for the definition of uniform continuity.
form an open (sub)cover of our space
The triangle inequality then implies that implying that
Applying the triangle inequality then yields the desired ∎ For an alternative proof in the case of
, a closed interval, see the article Non-standard calculus.