A continuous function $f : \mathbb{R} → \mathbb{R}$ is uniformly continuous if it maps Cauchy sequences into Cauchy sequences. A continuous function $f : \mathbb{R} → \mathbb{R}$ is uniformly continuous if it maps
Cauchy sequences into Cauchy sequences.
is the above statement is true?
I guess it is not true but can't find any counterexample.

The answer is no as explained by Jonas Meyer and every continuous function $f:\mathbb R\longrightarrow \mathbb R$ has this property:
If $(x_n)_{n\in\mathbb N}$ is a Cauchy sequence then $m\leq x_n \leq M, \ \ \forall \ n\in\mathbb N$ for some $m<M$. Since $f$ is uniformly continuous on $[m,M]$ the result follows. 
So $f(x)=x^2$ (or any continuous not uniformly continuous function $\mathbb R\to \mathbb R$) is a counterexample.
