Prove that the restriction $f\vert_A$ is continuous Let $S\subset \mathbb{R}$ and $A\subset S$. Let $f:S\rightarrow \mathbb{R}$ be a continuous function. Then the restriction $f\vert_A$ is continuous. 

By definition of the subspace topology, if $(X,\tau)$ is a topological space then $\{U\cap A | U\in\tau\}$ are the open sets of $A\subseteq X$. But then if $O$ is an open subset of $\Bbb R$ we have that
$$f\big|_A^{-1}(O) = f^{-1}(O)\cap A$$
but since $f$ is continuous, this is the intersection of open sets, hence is itself open, showing the restriction is continuous.
