54A20-ContinuousConvergence.tex

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\pmtitle{continuous convergence}
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Let $(X,d)$ and $(Y,\rho)$ be metric spaces, and let $f_n:X\longrightarrow Y$ be a sequence of functions. We say that $f_n$ \textit{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 $f_n$ \emph{converges continuously} to $f$ if it does for every $x \in X$.
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