The following Leibniz integral rule is well-known

Theorem. Let be a function such that both and its partial derivative are continuous in and in some region of the -plane, including , . Also suppose that the functions and are both continuous and both have continuous derivatives for . Then, for ,

The purpose of this note is to show that, in fact, it is not is not necessary to assume the function to be continuous. We note that this is indeed the case in which the limits of the integral do not depend on the parameter . For convenience, it is routine to assume the continuity, which immediately implies that all integrals are well-defined.

As mentioned above, we want to show that this is also the case for integrals of the form above.