 Rounded Bernstein polynomial

The rounded Bernstein polynomial replaces each coefficient $\left(\genfrac{}{}{0ex}{}{n}{k}\right)f(k/n)$ in the unnormalized basis $x^k(1-x{)}^{n-k}$ by its floor function. It therefore has integer coefficients in the ordinary monomial basis. If $f(0),f(1)\in \mathbb{Z}$, only interior rounding errors remain andTo prove this, bound each interior rounding error by one. On $[0,1/4]$, sum the resulting powers as a geometric series of ratio at most $1/3$ and maximize $x(1-x{)}^{n-1}$; reflect for $[3/4,1]$. On the middle interval each degree-$n$ product is at most $(3/4{)}^n$. integer endpoints are necessary for this convergence: a noninteger endpoint produces a fixed nonzero rounding error there.