Let X and Y be topological spaces, and f a mapping of X into Y. Show that f is continuous iff it is continuous

Let X and Y be topological spaces, and f a mapping of X into Y. Show that f is continuous iff it is continuous as a mapping of X onto the subspace f(X) of Y
thanks for your help

This question follows directly from the definitions. It is important that you can do these type of proofs.
I will do 1/2 of the problem. I claim that it takes no thought. Really.
Suppose f: X -> Y is continuous.
To prove: f: X -> f(X) is continous.
Let U be an open set in f(X).
By definition of the topology on f(X), U = V intersect f(X), where V is open in Y.
By definition of the continuity, f^-1 (V) is open in X.
By definition of f, f^-1(V) = f^-1(U).
By definition, this shows f : X->f(X) is continuous.