Prove $f$ is uniformly continuous iff $ \lim_{x\to \infty}f(x)=0$
813
$f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$
57642
Uniform Continuity implies Continuity
35325
Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.
121217
How to show that $\lim\limits_{x \to \infty} f'(x) = 0$ implies $\lim\limits_{x \to \infty} \frac{f(x)}{x} = 0$?
31741
Show for $f:A \to Y$ uniformly continuous exists a unique extension to $\overline{A}$, which is uniformly continuous
15727
show that if $f$ is uniformly differentiable then prove that $f'$ is continuous
9614
How does the existence of a limit imply that a function is uniformly continuous
57180
Techniques to prove a function is uniformly continuous
28730
product of two uniformly continuous functions is uniformly continuous
13257
How to prove this problem about uniform continuity
100979
Uniform continuity of a function and cauchy sequences
15284
$f$ defined on $[1,\infty )$ is uniformly continuous. Then $\exists M>0$ s.t. $\frac{|f(x)|}{x}\le M$ for $x\ge 1$.
123981
Is it possible to write any bounded continuous function as a uniform limit of smooth functions
55368
$f$ is continuous, if $f_n$ continuous and $f_n\to f$ uniformly
154034
Suppose that $f(x)$ is continuous on $(0, \infty)$ such that for all $x > 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function.
111290
If $f$ continuous and $\lim_{x\to-\infty }f(x)=\lim_{x\to\infty }f(x)=+\infty $ then $f$ takes its minimum.
110146
I need to prove the continuity of $f(x)=\log x$ using a $\epsilon-\delta$ proof