Prove a set is closed 
Suppose $f\colon \mathbb R \to \mathbb R$ is a continuous function and $K$ is a closed subset of $\mathbb R$. Prove that the set $A = \{x \in \mathbb R : f(x) \in k\}$ is also closed.       

Could someone show me direction as I am lost?

The definition of a continuous function is : A function that inverse image of open (closed) sets are open (closed). 
Thus your set $A=f^{-1}(K)$ is closed.
