The main result is the following generalization of the Moreau-Rockafellar theorem on the additivity of the convex subdifferential: If \(f\) and \(\{f_n\}\) are proper lower semicontinuous functions on a Banach space \(X\), with dual \(X^*\), such that \(\sum^\infty_{n= 1}f_n\) pointwise converges to \(f\) on \(X\) then, for each \(x\in\text{int dom}(f)\), \[ \partial f(x)= \Biggl\{x^*\in X^*: x^*= \sum^\infty_{n= 1} x^*_n\text{ (in the weak\(^*\) topology), with }x^*_n\in\partial f_n(x)\quad \forall n\Biggr\}. \] As a consequence, a generalization of the Kuhn-Tucker theorem, for convex optimization problems with a countable number of inequality constraints, is obtained. Other versions of the Moreau-Rockafellar theorem, involving norm (instead of weak\(^*\)) convergence, are also given.