Uniform Continuity
Similar to continuous functions between topological spaces, which preserve topological properties, are the uniform continuous functions between uniform spaces, which preserve uniform properties. Uniform spaces with uniform maps form a category. An isomorphism between uniform spaces is called a uniform isomorphism.
A uniformly continuous function is defined as one where inverse images of entourages are again entourages, or equivalently, one where the inverse images of uniform covers are again uniform covers.
All uniformly continuous functions are continuous with respect to the induced topologies.
Read more about this topic: Uniform Space
Famous quotes containing the words uniform and/or continuity:
“When a uniform exercise of kindness to prisoners on our part has been returned by as uniform severity on the part of our enemies, you must excuse me for saying it is high time, by other lessons, to teach respect to the dictates of humanity; in such a case, retaliation becomes an act of benevolence.”
—Thomas Jefferson (1743–1826)
“Every generation rewrites the past. In easy times history is more or less of an ornamental art, but in times of danger we are driven to the written record by a pressing need to find answers to the riddles of today.... In times of change and danger when there is a quicksand of fear under men’s reasoning, a sense of continuity with generations gone before can stretch like a lifeline across the scary present and get us past that idiot delusion of the exceptional Now that blocks good thinking.”
—John Dos Passos (1896–1970)