Let $f(A) = A^{-1}$ for a bounded linear operator $A$. How do we prove that $f$ is continuous?

Question

It seems I need to refresh some operator theory as its been quite a while since I need it. In another question I asked, someone stated that the inversion operation in $L(X,Y)$ is continuous, where $L(X,Y)$ is the set of bounded linear operators from $X$ to $Y$.

So define $f$ such that $A \to f(A): = A^{-1}$ for $A$ in the set of invertible operators. How do we prove that $f$ is continuous?