The paper concerns the stability, in the sense of Hyers--Ulam, of the monomial functional equation \[ \Delta^n_y f(x)-n!f(y)=0, \] where \(x,y \in \mathbb{R}\), and \(f\) takes values in a Banach space \(B\). The stability of this equation has been already studied by \textit{L. Székelyhidi}, [C. R. Math. Acad. Sci., Soc. R. Can. 3, 63-67 (1981; Zbl 0475.39015)]. In the present paper the author studies the stability in a restricted domain. The main result is the following Theorem: Let \(B\) be a Banach space, \(f:\mathbb{R} \to B\) be a function and \(n \in \mathbb{N}\). If for a real number \(\delta \geq 0\) we have \[ \|\Delta^n_y f(x)-n!f(y)\|\leq \delta, \] (\(x \leq 0 \leq x+ny\)), then there exist a \(c \in \mathbb{R}\) and a monomial function \(g:\mathbb{R} \to B\) of degree \(n\) such that \[ \|f(x)-g(x)\|\leq c\delta, \qquad x \in \mathbb{R}. \] Furthermore, there is only one monomial function of degree \(n\), for which there exists a \(c \in \mathbb{R}\) with this property.