We know that a continuous $f$ defined on an open interval $(a,b)$ where $a,b$ are real numbers is uniformly continuous iff the end point limits of $f$ are finite i.e.the function have a continuous extension to $[a,b]$.Is there any similar result for the interval $(a,\infty)$ where $a\in \mathbb R$.
2026-03-29 03:28:29.1774754909
Criterion of uniform continuity for an interval of the form $(a,\infty)$.
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Yes and no. The condition that $f$ approaches some finite value on both ends does guarantee uniform continuity, but it's not an iff statement anymore.
Take $f(x)=\log x$ as an example of a function that goes off to infinity on one side but still is uniformly continuous for $a>0$.
But a function that is usually nicer, $f(x)=x^2$ shares the above property with the log but is not uniformly continuous on the interval. This shows that the conditions grow independent from each other.