If f(x) is continuous, can it be uniformly continuous?

Question

The question states: Assume that $f(x)$ is continuous on $(0,3)$. Each of these statements are either true or false.

I am having trouble proving that $f$ is uniformly continuous on $(0,3)$ and that $f$ is uniformly continuous on $(1,2)$. I know what it means to be uniform continuous on a closed interval and what not, but I am having trouble on getting Theorems to show that they are uniformly continuous.

Please use some basic theorems about continuous and uniform continuity and bounds on the interval if necessary.

Answer 1

The function $f$ is continuous on the interval $(0,3)$ then it's continuous on the compact $[1,2]$ so by the Heine-Cantor theorem $f$ is uniformly continuous on $[1,2]$.

Answer 2

• $f$ need not be uniformly continuous on $(0,3)$. Take for instance $x\mapsto \frac1x$.
• if it is continuous on $(0,3)$, $f$ is in particular continuous on $[1,2]$. So apply Heine-Cantor theorem to get that $f$ is uniformly continuous on $[1,2]$, hence on $(1,2)$.