Prove that $f(x)$ is uniformly continuous on $I$ if and only if the image of each Cauchy sequence under $f$ is also a Cauchy sequence.
51247
$f$ be a continuous function maps Cauchy into Cauchy. Is $f$ uniformly continuous?
100786
Does a continuous function take Cauchy sequences to Cauchy sequences?
41482
$f_n → f$ uniformly on $S$ and each $f_n$ is cont on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$.
126212
Proving uniform continuity on an interval
84785
Is $f(x)=x+\frac{x}{x+1}$ uniformly continuous on $(0,\infty)$
94142
Proving that product of two Cauchy sequences is Cauchy
13257
How to prove this problem about uniform continuity
40297
Proof of uniform limit of Continuous Functions
23617
Determine whether $f(x)={\sin x \over x}$ is uniformly continuous in $\mathbb R$
53627
Proving continuity of $f$
30965
$f$ is continuous on $X$ iff $f$ is continuous on every compact subset
74364
Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$
56567
Prove or disprove the uniform continuity of $f: \mathbb{C} \to \mathbb{C}, z \mapsto f(z)=\frac{z^2}{1+|z|}$
15579
Show that $f$ is uniformly continuous if limit exists
3856
Uniform Continuity of $x \sin x$
117502
Proving that a certain continuous function is surjective.
129073
If $f'$ is bounded, then $f$ is uniformly continuous