On the rates of pointwise convergence for Bernstein polynomials

Let $f$ be a real function defined on the interval $[0,1]$ which is constant on $(a,b)\subset [0,1]$, and let $B_{n}f$ be its associated $n$th Bernstein polynomial. We prove that, for any $x\in (a,b)$, $\mathrm{\mid }{B}_{n}f(x)-f(x)\mathrm{\mid }$ converges to $0$ as $n\to \mathrm{\infty }$ at an exponential rate of decay. Moreover, we show that this property is no longer true at the boundary of $(a,b)$. Finally, an extension to Bernstein-Kantorovich type operators is also provided