Limits and Continuity, Continuous Function Becomes Uniform

Continuous Function Becomes Uniform

This proof is similar to the one before. Let f be continuous on a closed bounded region R in n dimensions and suppose it is not uniform. Hence there is an ε that requires arbitrarily small values of δ to assure continuity. Cut R into pieces and retain the subregion that fails to be uniform. This subregion requires arbitrarily small values of δ for the pathological ε. Do this again and again, cutting R into ever smaller pieces.

As before, the sequence approaches a point p, which is in R. Since f is continuous at p, let δ define a neighborhood about p such that f(x) is within ε/2 of f(p). There is a small region containing p, entirely inside this δ neighborhood, that is continuous, but not uniform. In fact, uniformity fails for the aforementioned ε. However f(x) and f(y) differ by no more than ε when x and y are contained in this region. There is a δ that satisfies ε for this region, which is a contradiction. Therefore f is uniform on all of R.