Lipschitz condition and Leibniz rule

Main Question or Discussion Point

Hi,
I am reading a paper that states: "We note that if an integrable function satisfies the Lipschitz condition of order one, then differentiation and integration can be interchanged. This provides a more compact way to take the derivative. Consequently, in our proofs, if an integrable function satisfies the Lipschitz condition of order one, then we interchange the differentiation and integration when we take the derivative of the function. Otherwise, we use the Leibniz’ rule to take the derivative".

Do they mean that if a function is not Lipschitz continous (of order 1) then Leibniz rule can be used?

Thanks!