Please help me do the following.
Suppose that $f:\mathbb{R}^m\to\mathbb{R}$ satisfies two conditions:
(i) For each compact set $K$, $f(K)$ is compact.
(ii) For any nested decreasing sequence of compacts $(K_n)$, $$f\left(\bigcap K_n\right)=\bigcap f(K_n).$$ Prove that $f$ is continuous.
Property (ii) implies the following: If $(x_i)$ is a sequence in $\mathbb{R}^m$ converging to $x$ such that $f(x_1)=f(x_2)=\ldots$, then $f(x)=f(x_1)$. I think they are in fact equivalent, but I'm not sure.
Edit: Sorry for the confusion, the second condition is now corrected.