Ali Akbar Estaji - Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Abolghasem Karimi Feizabadi - Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
Mohammad Zarghani - Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.

 A topoframe, denoted by L_{ \tau},  is a pair (L, \tau) consisting of a frame L and a subframe \tau all of whose elements are complementary elements in L. In this paper, we define and study the notions of a \tau -real-continuous function on a frame L and the set of real continuous functions \mathcal{R}L_\tau as an f-ring. We show that \mathcal{R}L_{ \tau} is actually a generalization of the ring C(X) of all real-valued continuous functions on a completely regular Hausdorff space X. In addition, we show that \mathcal{R}L_{ \tau} is isomorphic to a sub-f-ring of \mathcal{R}\tau . Let {\tau} be a topoframe on a frame L. The frame map \alpha\in\mathcal{R}\tau is called L-{\it extendable} real continuous function if and only if for every r\in \mathbb{R}, \bigvee^{L}_{r\in \mathbb R} (\alpha(-,r)\vee\alpha(r,-))'=\top. Finally, we prove that \mathcal{R}^{L}{\tau}\cong \mathcal{R}L_{\tau} as f-rings, where \mathcal{R}^{L}{\tau} is the set all of L-extendable real continuous functions of \mathcal{R}\tau .

frame, Topoframe, Ring of real continuous functions, Archimedean ring, f-ring