Let \(D\subset{\mathbb C}\) be a domain and let \(f : D\to \mathbb C\) be a continuous function. For \(z\in D\), and \(0<\varepsilon<\text{dist} (z,\partial D)\) set \[ M_\varepsilon(z)=\left\{{{f(z+h)-f(z)}\over{h}} ;\;0<|h|<\varepsilon\right\} \] and define the monogeneity set \(M(z)=\cap_{0<\varepsilon<\text{dist} (z,\partial D)}\overline{M_\varepsilon(z)}\). The authors prove the following sufficient condition for the analyticity of \(f\): Theorem. Let \(D\subset{\mathbb C}\) be a domain and let \(f : D\to\mathbb C\) be a continuous function which is monogenic in each everywhere dense subset \(E\) of \(D\). If \(f\) satisfies the condition that for any point \(\xi\in \mathbb C\) there are at most a countable family of sets \(M(z)\) containing \(\xi\), then \(f\) is a nonlinear analytic function on \(D\).