real analysis - Let $f:mathbb{R} rightarrowmathbb{R}$ such that $f$ is continuous and limit at $infty$ and $-infty $ exist then function is uniformly continuous?

Let $f:\mathbb{R} \rightarrow\mathbb{R}$ such that $f$ is continuous and limit at $\infty$ and $-\infty $ exist then function is uniformly continuous?

I do not think this is true as limit may exist at end points but what if function wiggle in between. May be I am wrong. Can I get some hint?

Answer

Let $L=lim_{x\rightarrow +\infty}f(x), L'=lim_{x\rightarrow -\infty}f(x)$, for every $c>0$, there exists $M>0$ such that $x>M$ implies that $|f(x)-L|

The function is uniformly continuous on $[-3M,3M]$ since $[-3M,3M]$ is compact, there exists $e$ such that for every $x,y\in [-3M,3M], |x-y|

Let $d=\inf(e,M)$. Consider $x,y\in \mathbb{R}$, such that $|x-y|M$ or $x,y<-M$, $|f(x)-f(y)|

Suppose that $x\in [-M,M]$ since $|x-y|