# Difference between Uniformally Continuous and Continuous

by kingstrick
Tags: continuous, difference, uniformally
 P: 828 Difference between Uniformally Continuous and Continuous OK, so let's say that I claim that $f$ is continuous on some interval, $I$, let's say the open unit interval. Then if you give me an $x_0 \in I$ and an $\epsilon > 0$ I should be able to give you a $\delta$ such that $|f(x_0) - f(x)| < \epsilon$ whenever $|x_0 - x| < \epsilon$. Now, it is important to note that I get to "see" $\epsilon$ AND $x_0$ before I have to come up with $\delta$. Now, if I claim that $f$ is uniformly continuous on some interval, $I$, say, the closed unit interval, then given an $\epsilon > 0$ I must come up with a $/delta$ that will work FOR ALL $x \in I$. That is, I don't get to "see" $x_0$ before I come up with $\delta$. So, why is this important? Well, if $I$ is compact and $f$ is continuous on $I$ then $f$ is uniformly continuous on $I$. Many of the theorems about derivatives, integrals, approximation of functions, etc, in analysis (at least at the undergrad level, which is all I know) require that the function be continuous on some closed and bounded (and hence compact) interval. This uniform continuity can then used to prove whatever is being proven.