Continuous and Uniformly Continuous Functions

We outline the difference between "point-wise" continuous functions and uniformly continuous functions. Basically, with "normal" or "point-wise" continuity, for any given point, for every ε, we can find a δ such that if the points are within δ, the imges are within ε. However, with uniform continuity, given some ε, we have to find a δ that works for every point of the curve. In other words, we pick the point first for normal continuity, and we pick the points after we pick ε and δ for uniform continuity. Another way of thinking about this is that point-wise continuity is a "local" condition , while uniform continuity is a "global" condition.

I try to illustrate this the best I can through examples and visual demonstration.