Combination_of_Recursive_Functions

Let $f: \N^k \to \N$ and $g: \N^k \to \N$ be recursive functions (not necessarily total), where $k \ge 1$. Let $\mathcal R$ be a $k$-ary relation such that if $\mathcal R \left({n_1, n_2, \ldots, n_k}\right)$ holds, then $f \left({n_1, n_2, \ldots, n_k}\right)$ is defined if $\mathcal R \left({n_1, n_2, \ldots, n_k}\right)$ does not hold, then $g \left({n_1, n_2, \ldots, n_k}\right)$ is defined. Let $h: \N^k \to \N$ be the function defined as $h \left({n_1, n_2, \ldots, n_k}\right) = \begin{cases} f \left({n_1, n_2, \ldots, n_k}\right) & : \text{if } \mathcal R \left({n_1, n_2, \ldots, n_k}\right) \text { holds} \\ g \left({n_1, n_2, \ldots, n_k}\right) & : \text{otherwise} \end{cases}$ so that $h$ is total. Then $h$ is recursive.