Let \(X\) be a locally Noetherian scheme, for example a variety, let \(p\in X\), and let \((\OO_{X,p},\mfm_p)\) be the local ring at \(p\). For \(f\in\OO_{X,p}\), define the order of vanishing of \(f\) at \(p\) as \[ \ord_p(f)\coloneqq\sup\theset{n\ge0\suchthat f\in\mfm_p^n}\in\ZZ_{\ge0}\cup\theset{\infty}. \] Then \(\ord_p(f)=\infty\) if and only if \(f=0\): a nonzero function vanishes to finite order at \(p\).
The local ring \(\OO_{X,p}\) is Noetherian, and \(\ord_p(f)=\infty\) means \(f\in\bigcap_{n\ge0}\mfm_p^n\), which is \(0\) by Krull's intersection theorem.
Nakayama's lemma for a local ring gives the companion statement for generators. Let \(k\) be a field and let \(\mfm_0\subseteq\OO_{\AA^n_k,0}\) be the maximal ideal at the origin. Functions \(f_1,\ldots,f_n\in\mfm_0\) generate \(\mfm_0\), that is, form a local coordinate system at \(0\), if and only if their classes form a basis of the cotangent space \(\mfm_0/\mfm_0^2\). The class of \(f\) in \(\mfm_0/\mfm_0^2\) is its linear part, the differential \(df_0=\sum_i\frac{\partial f}{\partial x_i}(0)\,dx_i\), so the condition is that the Jacobian matrix \(\qty{\frac{\partial f_i}{\partial x_j}(0)}\) is invertible, as in the inverse function theorem.