Characterize the continuous functions with finite right-hand derivative for at least one point of $[0, 1]$

Let $(E,d_\infty)$ the statistics room of continual features specified on $ [0,1] $, with $$ d_\infty(f,g)=\sup\{ |f(x) - g(x)| : x\in [0,1] \}. $$ For all $ n\in \mathbb{N} $ allow $$ F_n = \{ f: \exists x_0\in [0,1-1/n] \forall x\in [x_0,1]\left(|f(x)-f(x_0)|\leq n(x-x_0)\right) \}. $$ Let $D$ the set of continual features which have a limited derivate on the right for at the very least one factor of $[0, 1[$. I require to show that $$ D = \bigcup_{n\in\mathbb{N}} F_n. $$ This becomes part of trouble 38 of area 8 of Royden is Real Analysis publication.

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2022-06-07 14:38:29
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