Limits and Continuity, Uniform Continuity

Uniform Continuity

A function f is uniformly continuous, or uniform, throughout a region R if one δ fits all. In other words, δ depends on ε, but does not depend on the point p in the domain.

Every uniform function is continuous, but the converse is not true. Let f(x) = 1/x on the open interval (0,1). Choose δ small, as small as you like, and consider the image of the points in the interval (0,δ). Points in this set are arbitrarily far apart as they climb the y axis, hence they are not within ε of anything. This function, and any other function that approaches infinity, is not uniform.