# Definition(s) > [!NOTE] Definition (Pointwise convergence of Sequence of Real Functions) > Let $(f_{n})_{n=1}^\infty$ be a [[Sequences|sequence]] of [[Real Function|real functions]] from $\Omega \subset \mathbb{R} \to \mathbb{R}.$ We say that $(f_{n})$ or $f_{n}$ converges pointwise to $f:\Omega\to \mathbb{R},$ denoted $f_{n}\to f,$ iff for every $x\in \Omega,$ we have $ \lim_{ n \to \infty } f_{n}(x)\to f(x)$where $\lim_{ n \to \infty }f_{n}(x)$ denotes the [[Convergence|limit]] of the [[Real sequences|real sequence]] $(f_{n}(x))_{n=1}^\infty.$ > [!Example] Example > Consider the sequence $(f_{n})$ given by $f_{n}:[0,1]\to \mathbb{R},$ $f_{n}(x)=x^{\frac{1}{n}}.$ > > For all $x\in (0,1],$ $\lim_{ n \to \infty }x^{1/n} = 1.$ > > As a result limit of the sequence of real functions $(f_{n})$ is $f(x)=\begin{cases}0 & x=0, \\ 1 & x\in (0,1]. \end{cases}$ # Properties(s) ** # Application(s) **More examples**: # Bibliography