A function is analytic if it has a converging power series expansion. Please also use one of (real-analysis) or (complex-analysis) to specify real or complex analyticity: though the definition is the same, the properties of functions analytic over real numbers are quite different from the properties of functions over the complex numbers.
If $U$ is a subset of $\mathbb{C}$, $f$ is analytic at $x_0$ if there exists a series $$ \sum_{j=0}^\infty a_j (z-z_0)^j $$ that converges to $f$ at a neighbourhood of $z_0$. In complex analysis, analyticity is equivalent to holomorphy.
