Given a matrix $A \in \mathbb{R}^{n\times m}$, and its perturbation 
$$
A_p = A + \Delta
$$
is there a way to represent 
$$
(A_p)^{\star}= (A)^{\star} + f(\Delta)
$$
where $(A_p)^{\star}$ ($(A)^{\star}$) is the pseudo-inverse of $A_p$ ($A$)?
What can be said about the spectral norm of $f(\Delta)$?