 If formula_1 is a connected set, then the constant functions are the only antiderivatives of the zero function.Otherwise, a function is an antiderivative of the zero function if and only if it is constant on each connected component of formula_1 (those constants need not be equal).This observation implies that if a function formula_8 has an antiderivative, then that antiderivative is unique up to addition of a function which is constant on each connected component of formula_1.