Let $(E,d_\infty)$ the metric space of continuous functions defined on $ [0,1] $, with $ d_\infty(f,g)=\sup\{ |f(x) - g(x)| : x\in [0,1] \}. $ For all $ n\in \mathbb{N} $ let $ F_n = \{ f: \exists x_0\in [0,1-1/n] \forall x\in [x_0,1]\left(|f(x)-f(x_0)|\leq n(x-x_0)\right) \}. $ Let $D$ the set of continuous functions which have a finite derivate on the right for at least one point of $[0, 1[$. I need to show that $ D = \bigcup_{n\in\mathbb{N}} F_n. $ This is part of problem 38 of section 8 of Royden's Real Analysis book.
Characterize the continuous functions with finite right-hand derivative for at least one point of $[0, 1]$
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real-analysis
analysis
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0If $y$ou've found the answer, you should add it here, for future reference and other users. – 2012-10-27