Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), let $X$ be an irreducible affine variety over $k$, and let $k[X]$ be its coordinate ring.
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If $g,h\in k[X]$ are regular functions and there exists a nonempty open subset $U\subset X$ such that $g(p)=h(p)$ for every $p\in U$, then $g=h$ in $k[X]$.
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More generally, if $\varphi,\psi\in k(X)$ are rational functions and there exists a nonempty open subset $U\subset X$ contained in the common domain of definition of $\varphi$ and $\psi$ such that $\varphi(p)=\psi(p)$ for every $p\in U$, then $\varphi=\psi$ in $k(X)$.