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In mathematics, the uniform limit theorem states that the uniform limit of any sequence of continuous functions is continuous. ==Statement== More precisely, let ''X'' be a topological space, let ''Y'' be a metric space, and let ƒ''n'' : ''X'' → ''Y'' be a sequence of functions converging uniformly to a function ƒ : ''X'' → ''Y''. According to the uniform limit theorem, if each of the functions ƒ''n'' is continuous, then the limit ƒ must be continuous as well. This theorem does not hold if uniform convergence is replaced by pointwise convergence. For example, let ƒ''n'' : () → R be the sequence of functions ƒ''n''(''x'') = ''xn''. Then each function ƒ''n'' is continuous, but the sequence converges pointwise to the discontinuous function ƒ that is zero on [0, 1) but has ƒ(1) = 1. Another example is shown in the image to the right. In terms of function spaces, the uniform limit theorem says that the space ''C''(''X'', ''Y'') of all continuous functions from a topological space ''X'' to a metric space ''Y'' is a closed subset of ''YX'' under the uniform metric. In the case where ''Y'' is complete, it follows that ''C''(''X'', ''Y'') is itself a complete metric space. In particular, if ''Y'' is a Banach space, then ''C''(''X'', ''Y'') is itself a Banach space under the uniform norm. The uniform limit theorem also holds if continuity is replaced by uniform continuity. That is, if ''X'' and ''Y'' are metric spaces and ƒ''n'' : ''X'' → ''Y'' is a sequence of uniformly continuous functions converging uniformly to a function ƒ, then ƒ must be uniformly continuous. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Uniform limit theorem」の詳細全文を読む スポンサード リンク
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