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.