Show that this function is bounded Let $f$ be a $\mathbb R \rightarrow \mathbb R$ continuous function such that : $\lim_ {x \to \pm \infty} f(x) \in \mathbb R$ and $\lim_ {x \to 0} f(x) \in \mathbb R$
How can one show that $f$ is bounded ? I get it "intuitively" but I cant show it rigorously 

If $\lim_{x\to-\infty} f(x)=a$ and $\lim_{x\to\infty} f(x)=b$ put $|a|+|b|+1=:c$. There is an $M>0$ such that $|f(x)|\leq c
$ for all $x\geq M$ and all $x\leq-M$. Since $f$ is continuous there is a $c'$ such that $|f(x)|\leq c'$ for all $x\in[-M,M]$. It follows that $|f(x)|\leq c+c'$for all $x\in{\mathbb R}$.
