$f$ be a continuous function maps Cauchy into Cauchy. Is $f$ uniformly continuous?

Question

Let $f$ be a continuous function. If $f$ is uniformly continuous then it maps cauhy sequence into cauchy sequence. Is the converse true? That is if $f$ maps cauchy into cauchy gives the uniform continuity of $f$. I think it's no. Can suggest a counter example

Answer 1

Take $$f: \mathbb{R}\rightarrow\mathbb{R}$$$$f(x)=x^2$$

Let $(x_n)$ be a sequence of real numbers such that it is Cauchy. ie $(x_n)\rightarrow x$

Now $$f(x_n)=(x_n)^2=(x_n)(x_n)\rightarrow x\cdot x=x^2$$ Thus $f(x_n)$ is Cauchy, however you can show that $f(x)=x^2$ is not uniformly continuous