A multifunction \(F : X \to Y\) from a topological space into normed linear space \(Y\) is called restricted weak usc at \(x \in X\) if given a neighbourhood \(W\) of zero in \(Y\) endowed with the weak topology there exists a neighbourhood \(U\) of \(x\) such that \(F(U) \subset F(x) + W\). For weakly compact valued multifunctions such notion reduces to weak upper semicontinuity. The paper contains characterizations of restricted weak usc multifunctions, applications to subdifferentials, relations between restricted weak and strong (norm) usc, an interesting selection theorem for \(F\) defined on a \((\sigma - \beta)\)-unfavorable space, the generic continuity theorem exhibiting conditions under which a weakly usc multifunction is single-valued and strongly usc at the points of a residual subset of its domain. The results are applied to Fréchet differentiability of continuous convex functions and certain locally Lipschitzian functions.