Derivative of A Constant - Antiderivative of Zero

Antiderivative of Zero

A partial converse to this statement is the following:

If a function has a derivative of zero on an interval, it must be constant on that interval.

This is not a consequence of the original statement, but follows from the mean value theorem. It can be generalized to the statement that

If two functions have the same derivative on an interval, they must differ by a constant,

or

If g is an antiderivative of f on an interval, then all antiderivatives of ƒ on that interval are of the form g(x) + C, where C is a constant.

From this follows a weak version of the second fundamental theorem of calculus: if ƒ is continuous on and ƒ = g' for some function g, then