This concept addresses the theory of antiderivatives for functions with a discontinuity, showing that when a function's domain is split by a discontinuity, its antiderivative family gains independent constants of integration on each side of the discontinuity rather than a single shared constant. This occurs because the standard theorem — that two functions sharing a derivative on an interval differ by a single constant — relies on the Mean Value Theorem, whose continuity/differentiability hypotheses fail across a discontinuity, so the theorem's conclusion (and the usual "+C" convention) does not extend across it. This belongs to single-variable calculus, specifically the theory of antiderivatives and integration constants, and has direct consequences for classic results such as the antiderivative of 1/x being ln|x| + C, where the constant must be understood as potentially different on the positive and negative branches.
This concept addresses the theory of antiderivatives for functions with a discontinuity, showing that when a function's domain is split by a discontinuity, its antiderivative family gains independent…