In this paper we extend some results regarding the properties of weak upper gradients, from the cases when B is an Orlicz space or a Lorentz space to the general case of a Banach function space. We provide methods to cut and paste B-weak upper gradients and give extensions to the case of B-weak upper gradients for the product rule and the chain rule. These results require no additional assumptions on the Banach function space B. We also prove the existence of a norm minimizing B-weak upper gradient for a function possessing at least one B -weak upper gradient that belongs to B,under the assumption that B is reflexive or B has an absolutely continuous norm. If in addition the norm of B is strictly monotone, it turns out that a norm minimizing B-weak upper gradient of a function is also minimal pointwise Ό-almost everywhere among all the B-weak upper gradients of that function.
