Let $f$ be a real-valued continuous function defined on the interval $[0,1]$, and let $B_{n,\lambda }(f;\cdot )$ be the associated $n$-th degenerate Bernstein polynomial. We proved that, on any admissible interval, $|B_{n,\lambda }(f;\cdot )-f(\cdot )|$ converges uniformly to $0$ as $n\to \mathrm{\infty }$. We also established the uniform convergence of the first derivative of the degenerate Bernstein polynomial. To support the theoretical results, we tested the approximation theory on $f(x)=\sin (\pi x)$ and its first-order derivative $f^{\prime }(x)=\pi \cos (\pi x)$. These showed that admissible pairs, and some formal pairs close to the admissible range, give better function approximation than the classical Bernstein case, and the first-derivative approximation convergence increases as ${\lambda }_n\to 0$.