## Real analysis Limits and continuous functions November 26 3

Suppose that $f:\mathbb{R}\to \mathbb{R}$ is continuous on $\mathbb{R}$ and that $$ \lim_{x\to -\infty} f(x)=\lim_{x\to +\infty} f(x) =k$$ Prove that $f$ is bounded and if there exist a point $x_0 \in\mathbb{R}$ such that $f(x_0)>k$, then $f$ attains