The theory of focal points and conjugate points is an important part of the study of problems in the calculus of variations and control theory. In previous works we gave a theory of focal points and of focal intervals for an elliptic form
J
(
x
)
J(x)
on a Hilbert space
A
\mathcal {A}
. These results were based upon inequalities dealing with the indices
s
(
σ
)
s(\sigma )
and
n
(
σ
)
n(\sigma )
of the elliptic form
J
(
x
;
σ
)
J(x;\sigma )
defined on the closed subspace
A
(
σ
)
\mathcal {A}(\sigma )
of
A
\mathcal {A}
, where
σ
\sigma
belongs to the metric space
(
Σ
,
ρ
)
(\Sigma ,\rho )
. In this paper we give an approximation theory for focal point and focal interval problems. Our results are based upon inequalities dealing with the indices
s
(
μ
)
s(\mu )
and
u
(
μ
)
u(\mu )
, where
μ
\mu
belongs to the metric space
(
M
,
d
)
,
M
=
E
1
×
Σ
(M,d),M = {E^1} \times \Sigma
. For the usual focal point problems we show that
λ
n
(
σ
)
{\lambda _n}(\sigma )
, the nth focal point, is a
ρ
\rho
continuous function of
σ
\sigma
. For the focal interval case we give sufficient hypotheses so that the number of focal intervals is a local minimum at
σ
0
{\sigma _0}
in
Σ
\Sigma
. Neither of these results seems to have been published before (under any setting) in the literature. For completeness an example is given for quadratic problems in a control theory setting.