# continuous convergence

Let $(X,d)$ and $(Y,\rho)$ be metric spaces, and let $f_{n}:X\longrightarrow Y$ be a sequence of functions. We say that $f_{n}$ converges continuously to $f$ at $x$ if $f_{n}(x_{n})\longrightarrow f(x)$ for every sequence $(x_{n})_{n}\subset X$ such that $x_{n}\longrightarrow x\in X$. We say that $f_{n}$ converges continuously to $f$ if it does for every $x\in X$.

Title continuous convergence ContinuousConvergence 2013-03-22 14:04:58 2013-03-22 14:04:58 Mathprof (13753) Mathprof (13753) 7 Mathprof (13753) Definition msc 54A20 converges continuously