What is uniform continuity geometrically

Question

I just wonder if someone can give me clear picture because I do understand the technicalities but I still don't have a clear picture of the difference in the geometry between a uniform continuous function and a continuous function.

Answer 1

Basically, continuity is a local property, while uniform continuity is global.

Informally: Say $f$ takes value $y$ at point $x$. Now you perturb the argument $x$ "a little" ($\pm\delta$). Then the values of $f$ around $x$ deviate only by "a little" $\pm\epsilon$ from $y$. For every $\epsilon$ you find such a $\delta$ around $x$. This is continuity at $x$ and it's local, so a continuous function just satisfies this condition independently at every $x$. In contrast, if $f$ is uniformly continuous, the choice of $\delta$ depends only on $\epsilon$, but not on the point: the same $\delta$ works throughout the domain. So given $\epsilon >0$, consider the tunnel given by the graphs of $f\pm\epsilon$. There now exists $\delta>0$ with the property that if you shift the graph of $f$ horizontally by less than $\delta$, then the complete shifted graph is still in the tunnel - this isn't true for merely continuous $f$.