$D\subset \mathbb{R}$ and $f:D \to \mathbb{R} $
(1) For every pair of sequences $\{u_n\},\{v_n\}$ in $D$ satisfying $\lim|u_{n}-v_{n}|=0 $,$\lim|f(u_n)-f(v_n)|=0$ holds. $\implies $ $f$ is uniformly continuous.
(2) if $\{x_n\}$ is Cauchy in $D$ ,then $\{f(x_n)\}$ is a Cauchy sequence in $\mathbb{R}$. $\implies $ $f$ is uniformly continuous.
I know inverse implications of these are true. But, are these implications true ?
Every continuous map from $\Bbb R$ into $\Bbb R$ maps Cauchy sequences into Cauchy sequences. Since there are continuous map from $\Bbb R$ into $\Bbb R$ which are not uniformly continuous…
On the other hand, if $f\colon D\longrightarrow\Bbb R$ is not uniformly continuous, then there is some $\varepsilon>0$ such that, for each $\delta>0$, there are numbers $x,y\in D$ such that $|x-y|<\delta$ and $\bigl|f(x)-f(y)\bigr|\geqslant\varepsilon$. So, for each $N\in\Bbb N$, take $x_n,y_n\in D$ such that $|x_n-y_n|<\frac1n$ and that $\bigl|f(x_n)-f(y_n)\bigr|\geqslant\varepsilon$. Then $\lim_{n\to\infty}|x_n-y_n|=0$, but you don't have $\lim_{n\to\infty}\bigl|f(x_n)-f(y_n)\bigr|=0$.