A sufficient condition under which a continuous function on an *unbounded open interval* $I$ will be uniformly continuous on $I$

Question

*A sufficient condition under which a continuous function on an unbounded open interval $I$ will be uniformly continuous on $I$*

If the the interval $I$ is closed and bounded, then the continuous function is uniformly continuous.

But what about the case when the interval is unbounded open interval .so is there any sufficient condition for that.

Answer 1

Some conditions are:

1.If $f$ is Lipschitz continuous then is uniformly continuous.

2 If $f:(a,+\infty) \to \Bbb{R}$ continuous and $\lim_{x \to a^+}f(x),\lim_{x \to +\infty}f(x) \in \Bbb{R}$ then the function is uniformly continuous. Similarly for intervals of the form $(-\infty,a)$.

3.If $f:\Bbb{R} \to \Bbb{R}$ continuous and $\lim_{x \to +\infty}f(x)=\lim_{x \to -\infty}f(x)=0$ then $f$ is uniformly continuous.

4.If $f: \Bbb{R} \to \Bbb{R}$ is continuous and periodic then it is uniformply continuous.

5.If $f: \Bbb{R} \to \Bbb{R}$ is continuous,monotone and bounded,then it is uniformly continuous

6.If $f: \Bbb{R} \to \Bbb{R}$ is continuous at zero and $f(x)+f(y)=f(x+y),\forall x,y \in \Bbb{R}$,then $f$ is uniformly continuous.

Answer 2

If the function f has first order bounded derivative on the open unbounded interval I ; then f is uniformly continuous on I.