Uniform Continuity
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Recall that the definition of continuity is the function limit
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In our discussion of function limits we said that what we are attempting to do is find a <Sign in to see all the formulas> that is dependent on <Sign in to see all the formulas>. In other words, find a <Sign in to see all the formulas> so that (now we are just repeating the definition) for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that if <Sign in to see all the formulas>, then <Sign in to see all the formulas>.
But here is the point that is very important. What we did not emphasize at the time was that <Sign in to see all the formulas> is also dependent on $a$ as well! For example, when we proved the continuity of $x^2$, we found <Sign in to see all the formulas>. We really should have denoted it by something like <Sign in to see all the formulas>. But now that we know what we have done, wouldn't it be nice to have a definition of continuity that was not dependent on $a$? In other words, find a <Sign in to see all the formulas> that could work for any value of $a$. We call this Uniform Continuity and here is its definition:
| Uniform Continuity |
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| A function $f(x)$ is said to be uniformly continuous on some subset of the x-axis, which we denote by $U$, if for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that for every $x$ and $z$ in $U$, if <Sign in to see all the formulas>, then <Sign in to see all the formulas>. |
Finally, there are three important things to note about uniform continuity:
- If $f(x)$ is uniformly continuous on a set $U$ and $a$ is inside $U$, then $f(x)$ is continuous at $a$.
- The straight line is a function that is continuous for every $a$ inside $U$ and also uniformly continuous on $U$. For a straight line $U$ is the entire $x$-axis. See here and here. However, this is a specific example that does not indicate how things work in general. Just because a function is continuous for every $a$ inside $U$ does not guarantee that it is uniformly continuous on $U$. This is a very subtle point best demonstrated by example. For example, we have proven here that $x^2$ is continuous on the entire $x$-axis. However, unlike a straight line, we can prove that it is not uniformly continuous on the entire $x$-axis.
- Finally, there is one situation where being continuous implies uniform continuity. If $f(x)$ is continuous on the closed interval $[a,b]$, then $f(x)$ is uniformly continuous on $[a,b]$. This result is very hard to prove but very important. The main use of this result is to prove that continuous functions are integrable.
