Undoubtedly quite simple, but I'm stuck:
If $f$: $[a, b)\longrightarrow\mathbb{R}$ is uniformly continuous, must $\lim\limits_{x\longrightarrow b}f(x)$ exist?
2026-04-01 09:51:27.1775037087
Uniform continuity question
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The answer is yes. By definition of uniformly continuous, for any $\epsilon>0$, there is a $\delta>0$ such that for any $x,y\in(b-\delta,b)$, it is always true that $|f(x)-f(y)|<\epsilon$. By Cauchy criterion, this means that $\lim\limits_{x\to b}f(x)$ exist.