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.