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. 

You are saying that $f:S\to\mathbb R$ is continuous. This implicates that $S$ and $\mathbb R$ are both equipped with a topology.

The restriction can be written as:$$f\circ i_A$$ where $i_A:A\to S$ denotes the inclusion function prescribed by $a\mapsto a$.
If $A\subset S$ is equipped with the subspace topology inherited from $S$ then $i_A$ is continuous, hence composition $f\circ i_A$ is continuous.
