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

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.

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Hint: the point with finite right derivative is $x_0$. – Yuval Filmus May 22 '11 at 21:49
This looks wrong to me. Are you sure you want $x_{0} \in [0,1/n]$ and not $x_{0} \in [0,1-1/n]$? Then you should be able to apply the hint given by @Yuval. – t.b. May 22 '11 at 21:53
Yes, that is the point @Yuval. And You are right @Theo, thank you. – leo May 22 '11 at 22:05
You should probably draw a picture. You know that the graph lies below some line on the interval $[x_0, x_0 + \delta]$. Now you also know that the function attains its maximum $M$ somewhere. If the line intersects the vertical line at $x_{0} + \delta$ at a higher point than $M$ then you're done. If not, simply make the line steeper to achieve that. – t.b. May 22 '11 at 22:23
Whoa, wait a minute! I only commented on the very first comment with a formula. I didn't see the others (I had a coffee in the meantime and left this page open) and I haven't read them. It seems to me that it should be much simpler. – t.b. May 22 '11 at 22:25