Some necessary conditions for having nonempty weak subdifferential of a function are presented and the positively homogeneous of the weak subdifferential operator is proved. Necessary and sufficient conditions for achieving a global minimum of a weak subdifferentiable function is stated, as well as a link between subdifferential and the Fréchet differential with a weak subdifferential. A result about the equality of the fuzzy sum rule inclusion is also investigated. Finally, some examples are included.