Regular function

Remarks

In case 2, $g/h$ is a well-defined function on $U$, because $g$ and $h$ have the same degree $d$ and so $g(\lambda a)/h(\lambda a) = g(a)/h(a)$. A regular function is continuous when $k$ is identified with $\mathbb{A}^1$ with its Zariski topology.

Example: on $X = k \setminus \set{0}$, the function $f(x) = 1/x$ is regular, with $g = 1$ and $h = x$ on $U = X$.

Non-example: the function on $k$ given by $f(0) = 1$ and $f(x) = 0$ for $x \ne 0$ is not regular at $0$. Near $0$ it would equal $g/h$ on a nonempty open set, so $g$ would vanish on a cofinite set and hence be $0$, which contradicts $f(0) = 1$.

On an affine variety $X$, the ring of regular functions on all of $X$ equals the coordinate ring $k[X]$; this uses Hilbert's Nullstellensatz. On a projective variety, every regular function on the whole variety is constant.