The main theorem of the paper says that if a function
from a real normed linear space
into an Abelian group
satisfies the conditional Cauchy equation
is additive. This result generalizes some theorems of C. Alsina
and J. L. Garcia
[On a conditional Cauchy equation on rhombuses (to appear)] and its proof contains an interesting connectivity theorem. An application to a class of orthogonal additive mappings is given.