Integer-coefficient polynomial approximation (${\overline{\mathbb{Z}[x]}}^{\Vert \cdot {\Vert }_{\infty }}=\{f\in C[0,1]:f(0),f(1)\in \mathbb{Z}\}$)

The uniform approximation closure of the integer-coefficient polynomials on $[0,1]$ consists precisely of the continuous functions taking integer values at both endpoints. Necessity follows because evaluations at zero and one are integers and convergent integer sequences are eventually constant. For sufficiency, the rounded Bernstein polynomial has integer monomial coefficients and differs uniformly by a vanishing amount from the Bernstein polynomial, which converges uniformly to the target.