Checking uniform continuity of functions defined on the interval $(0, \infty)$

I know that a continuous function $f:[0, \infty) \to \mathbb{R}$ is uniformly continuous if $\lim_{x \to \infty} f(x)$ exists in $\mathbb{R}$.

But what can we say about functions on $(0,\infty)$ ?

Are there any rules/ techniques which can be used to predict uniform continuity if the $\epsilon - \delta$ method doesn't seem to work ?

I know three useful ways (till now) to handle such a situtaion.

• Every periodic continuous function is uniformly continuous.
• Any function having a bounded derivative is uniformly continuous.
• $f : X \to Y$ is uniformly continuous. $\Longleftrightarrow$ $\forall$ $\{a_n\}$ and $\{b_n\}$ in $X$ such that $d(a_n,b_n) \to 0$, we have $d(f(a_n),f(b_n)) \to 0$.

But, are there any other ways to decide ?

Any input/ advice is welcome.

Solution 1:

If both $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to0}f(x)$ exist, then it is uniformly continuous on $(0,\infty)$.

We know that a continuous function defined on a closed interval is uniform continuous. So if it is not uniform continuous, the problem must occur around the boundaries. There are two possible cases for a continuous function to be not uniform continuous when it is close to its boundary ($\infty$ can also be viewed as a general boundary): 1) It tends to infinity. 2) It oscillates vastly. As long as these two situations don't happen, you will obtain a uniformly continuous function.