The differences for properties of the range space/( A') are considered for functions f: X -Y, where / is almost continuous relative to X X Z, and /( X) C Z C Y. It is shown that if / is allowed to be almost continuous relative to X X Z, where A' is a Peano continuum and Z is a locally connected metric space, then /( X) can be any type of subcontinuum of Z. This contrasts the known results for the case where Z=f(X) and almost continuity is relative to XXf(X). Outer almost continuous retracts (f:

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... X -X is almost continuous relative to XXX) and inner almost continuous retracts ( /: X -» X is almost continuous relative to X X /( X)) are defined. Properties of outer almost continuous retracts, including the existence of an outer almost continuous retract M of a fixed point space X, where M does not have the fixed point property, are found.