The ring of real-continuous functions on a topoframe

 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 $alphainmathcal{R}tau $ is called $L$-{it extendable} real continuous function if and only if for every $rin mathbb{R}$, $bigvee^{L}_{rin mathbb R} (alpha(-,r)veealpha(r,-))'=top.$ 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 $.