# Definition A real [[Map|function]] of a single real variable, $f(x)$ is said to be *analytic* at $x = x_0$ if it has a Taylor expansion about $x_0$ that converges to $f(x)$ in some [[Neighborhood|neighborhood]] of $x_0$: $f(x) = f(x_0) + (x-x_0)\left(\frac{df}{dx}\right) + \frac{1}{2}(x-x_0)\left(\frac{d^2f}{dx^2}\right)+ \ldots$ Naturally, a function needs to be infinitely [[Differentiable Function|differentiable]] at $x = x_0$ for it to be analytic, but not all infinitely [[Differentiable Function]] are analytic. Analytic functions are denoted by $C^\omega$.