What is an example of a function that is uniformly continuous but not continuous on an interval $[a,b]$?

Question

In the Appendix to Chapter 8 of Spivak's Calculus, entitled "Uniform Continuity", there is the following theorem

If $f$ is continuous on $[a,b]$, then $f$ is uniformly continuous on $[a,b]$

The absence of the converse seems to imply that it isn't true. What is an example of a function that is uniformly continuous but not continuous on an interval $[a,b]$?

Answer 1

The absence of the converse seems to imply that it isn't true.

Not really. The absence of the converse sometimes means that it isn't true. Other times, it means that the converse is more broadly true.

In this case, the converse is more broadly true. In fact, if $I$ is a bounded interval, and $f: I\to\mathbb R$ is uniformly continuous, then $f$ is continuous on $I$. The converse is only true if $I$ is a closed interval.

In other words,

• if $f$ is uniformly continuous on $(a,b)$, then $f$ is continuous on $(a,b)$.
• if $f$ is uniformly continuous on $[a,b]$, then $f$ is continuous on $[a,b]$.
• if $f$ is continuous on $[a,b]$, then $f$ is uniformly continuous on $[a,b]$.
• if $f$ is continuous on $(a,b)$, then we cannot say whether $f$ is uniformly continuous on $(a,b)$. That is, it might be, but we cannot conclude that just because it is continuous on $(a,b)$, that it them must be uniformly continuous on $(a,b)$. For example, $x \mapsto \frac1x$ is continuous, but not uniformly continuous, on $(0,1)$.