Let \(X\) be a Banach space and let \(L(X)\) be the space of all linear continuous operators on \(X\). For \(T \in L(X)\), let \(\mathcal A_T\) denote the closure of the algebra generated by \(T\) and the identity operator \(I\) in the weak operator topology. The celebrated von Neumann Double Commutant Theorem states that, if \(X\) is a Hilbert space and \(\mathcal A \subseteq L(X)\) is a weakly closed selfadjoint unital algebra of operators, then \(\mathcal A\) equals its double commutant \(\mathcal A^{cc}\). Several authors have investigated various extensions of this theorem. Let \(K\) be a metrizable, connected and locally connected compact space. Denote by \(C_{\mathbb R} (K)\) the space of all continuous real valued functions on \(K\). In the present paper, the author shows that, for every \(F \in C (K)\), \(\{M_F\}^{cc} = \mathcal A_F\), where \(M_F\) is the multiplication operator corresponding to \(F\) on \(C_{\mathbb R} (K)\). The complex case is also discussed. The author also considers multiplication operators \(M_F\) on the space \(C^{(1)}_{\mathbb R} [0,1]\) of all continuously differentiable real valued functions on \([0,1]\) endowed with the norm \( \| f \|= |f (0)| + \| f'\|_{\infty}\). More specifically, he gives a complete description of polynomials (respectively, non-decreasing functions) \(F\) satisfying \( \{ M_F\} ^{cc} = \mathcal A_F\).