Metric Spaces, Uniform Continuity

Uniform Continuity

Adopt the δ ε definition of continuity, as described in the previous section. A function f is uniformly continuous, or uniform, throughout a region R if one δ fits all. In other words, δ is a function of 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.

If f and g are continuous, consider f(g()). Pull an ε neighborhood in f(g()) back to a δ neighborhood in g(), back to a γ neighborhood in the domain. Thus the compositionn of continuous functions is continuous. But we already knew that; we proved it using open sets. The new result is that the composition of uniform functions is uniform. Regardless of where we are in the domain, ε determines δ determines γ, hence the composition is uniform.