 Integer polynomial approximation away from zero and one

On $[a,b]\subset (0,1)$, the uniform closure of integer-coefficient polynomials contains $1/2$, since ${\sum }_{j=0}^{N}x(1-2x{)}^j\to 1/2$ uniformly. This closure is a ring, so it contains all dyadic constants and then all real constants. It contains $x$ and therefore every real-coefficient polynomial. The Stone-Weierstrass theorem makes it all of $C([a,b])$. On an interval including zero this fails, since the value of an integer-coefficient polynomial at zero is an integer.