Proving restriction of function is continuous The problem is:

If $f : (X, T) \to (Y, S) $ is continuous and $A \subseteq X$, then $f|_A : (A, T_A) \to(Y, S) $ is continuous.

How would I get started on this proof? I understand the restriction of the function $f$ is $f|_A(x) = f(x) $ for all $x\in A $. I jut don't see how to use this to show that it is continuous.

If $A$ has the subspace topology, then $f^{-1}(U)$  is open for any $U \subset Y$ that is open. But then $f^{-1}(U) \cap A$ is open in $A$, but $f^{-1}(U) \cap A$ is precisely$\dots$
