Show that $f$ is uniformly continuous given $\lim_{x \to -\infty}f(x)=a$ and $\lim_{x \to \infty}f(x)=b$
106060
If $f$ is continuous and with a limit at infinity then $f$ is uniformly continuous
12997
Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous?
93461
Suppose that $f:\Bbb R\to \Bbb R$ is continuous and that $f(x)\to 0$ as $x\to \pm\infty.$Prove that $f$ is uniformly continuous.
33304
Let $f$ be a continuous function satisfying $\lim \limits_{n \to \infty}f(x+n) = \infty$ for all $x$. Does $f$ satisfy $f(x) \to \infty$?
126212
Proving uniform continuity on an interval
127538
Show $f(x)=x\ln x$ is not uniformly continuous
113086
Prove or disprove that $f(x) = \sin x/x$ is uniformly continuous over the interval $(0,1)$?
129856
Show that $f(x)=\frac{x}{1+|x|}$ is uniformly continuous.
59162
Prove $\liminf\limits_{n\to\infty} F(x_n)\le F(x)$
124779
Intuition for uniform continuity of a function on $\mathbb{R}$
125024
Is this function uniformly continuous?
84785
Is $f(x)=x+\frac{x}{x+1}$ uniformly continuous on $(0,\infty)$
150554
proof that $f'$ is continuous at $a$
48891
Show that: lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$
29287
Prove that: $\lim_{n\to\infty} f(n+1) - f(n) = \lim_{x\to\infty} (f(x))' $
55605
If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
135700
Show that $\lim_{n \to \infty} (1/n)\sum_{i=1}^n f(x_i)$ exists