If $f: [1, \infty ) \rightarrow \mathbb{R}$ is uniformly continuous, prove that $f(x)$ is bounded by some positive, finite $M$ times $x$
15284
$f$ defined on $[1,\infty )$ is uniformly continuous. Then $\exists M>0$ s.t. $\frac{|f(x)|}{x}\le M$ for $x\ge 1$.
124779
Intuition for uniform continuity of a function on $\mathbb{R}$
102143
$f:\mathbb R\rightarrow \mathbb R$ be a continuous function such that $\int_{0}^{\infty}f(x)dx$ exists.
12408
Prove that a function whose derivative is bounded is uniformly continuous.
3518
Proving $f(x)=1/x$ on $(0,1 )$ is not uniformly continuous
65346
Characterisation of uniformly continuous function
15579
Show that $f$ is uniformly continuous if limit exists
72240
Prove that if a linear operator is continuous, then it is bounded.
143695
Is an increasing, bounded and continuous function on $[a,+\infty)$ uniformly continuous?
123981
Is it possible to write any bounded continuous function as a uniform limit of smooth functions
127538
Show $f(x)=x\ln x$ is not uniformly continuous
40297
Proof of uniform limit of Continuous Functions
76607
Show that exists a continuous function in $E$ that is not bounded
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.
139612
Follow-up regarding right-continuous $f:\mathbb{R} \to\mathbb{R}$ is Borel measurable
2118
Are continuous functions with compact support bounded?
55368
$f$ is continuous, if $f_n$ continuous and $f_n\to f$ uniformly