A constant added to a function changes its reference level without changing its derivative:Geometrically, it is a vertical translation of the graph. For , changing moves every plotted value by the same amount while preserving every tangent slope. This is the geometric reason that antiderivatives form a family.
When represents a physical quantity, the physical dimension of must match that of . When position is obtained from velocity, is a position offset. An unknown starting position remains unknown even when the rate of change is known exactly.
When describing the family of antiderivatives on an interval, the arbitrary additive constant is the constant of integration. On a disconnected open domain, the corresponding freedom is a locally constant function, as in antiderivatives on disconnected domains, rather than necessarily one constant on the whole domain.
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