The local magnetization of a spin at an arbitrary distance $(n-1)$ from the edge of the lattice is rigorously calculated for the semi-infinite two-dimensional Ising model. It is shown that as $T\to T_c$ , $n\to\infty$ the magnetization takes the scaling form $\langle s_n\rangle =\tau^{1/8}F(x)$ ( $\tau=|1-T/T_c|$ , $x\sim 2n \tau$ ). Exact expressions are found for the function $F(x)$ and its asymptotic behavior at large and small $x$ is found.