The differences for properties of the range space
f
(
X
)
f(X)
are considered for functions
f
:
X
→
Y
f:X \to Y
, where
f
f
is almost continuous relative to
X
×
Z
X \times Z
, and
f
(
X
)
⊂
Z
⊂
Y
f(X) \subset Z \subset Y
. It is shown that if
f
f
is allowed to be almost continuous relative to
X
×
Z
X \times Z
, where
X
X
is a Peano continuum and
Z
Z
is a locally connected metric space, then
f
(
X
)
f(X)
can be any type of subcontinuum of
Z
Z
. This contrasts the known results for the case where
Z
=
f
(
X
)
Z = f(X)
and almost continuity is relative to
X
×
f
(
X
)
X \times f(X)
. Outer almost continuous retracts (
f
:
X
→
X
f:X \to X
is almost continuous relative to
X
×
X
X \times X
) and inner almost continuous retracts (
f
:
X
→
X
f:X \to X
is almost continuous relative to
X
×
f
(
X
)
)
X \times f(X))
) are defined. Properties of outer almost continuous retracts, including the existence of an outer almost continuous retract
M
M
of a fixed point space
X
X
, where
M
M
does not have the fixed point property, are found.