In this section demonstrate continuity and Gateaux differentiability
of 
under our current assumptions on 
.
In subsequent subsections we show that, with additional
assumptions, Fréchet differentiability may be obtained.
As we are interested in inverse problems involving the determination of a particular q from knowledge or partial knowledge of F(q) we discuss the regularity of the map F. The following lemma, essentially due to Gutman [15], will be useful.
PROOF:
In preparation for considering 
we have, for 
,
so 
.
Also, 
implies
.
Thus it suffices to show that
,
implies
But since K is a compact subset of 
, the set
is compact in 
. The result
then follows directly from [15, Lemma 3.1,]. We shall also have need of the following simple formulae:
To show (23), we note that from
the linearity of 
we have
. Left-multiplying this equation
by 
and right-multiplying by 
yields (23). Equation (24)
follows similarly.
From (23) and Lemma 2.1, we can
easily establish the continuity of .
PROOF:
We need to show that for fixed 
,
as
.
But from the definition (21) of F, the inverse
perturbation formula (23), and the
bound (19), we have
Letting K be the singleton 
in
Lemma 2.1, we obtain the result.
