User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

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Theorem
Let $\map \BB {\R, \size {\, \cdot \,} }$ be the Borel Sigma-Algebra on $\R$ with the Usual Topology.

Let $f: \R \to \R$ be a Continuous Real Function.

Let $\map D f$ be the set of all points at which $f$ is differentiable.

Then $\map D f$ is a Borel Set with respect to $\map \BB {\R, \size {\, \cdot \,} }$.

Proof
By the definition of derivative:


 * $\map {f'} x$ exists




 * $\displaystyle \lim_{h \mathop \to 0} \frac{\map f {x+h}-\map f x }{h} - L$ exists

where $L = \map{f'}{x}$.

By the definition of limit, this statement is:


 * $\displaystyle \exists L \in \R: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall h \in \R \setminus \set{0}: \size{h}<\delta \implies \size{\frac{\map f {x+h}-\map f x}{h} -L} < \epsilon$.

From Limit with Rational $\epsilon$ and $\delta$ and Limit with $\epsilon$ Powers of $2$, we can instead consider:


 * $\displaystyle \exists L \in \R: \forall n \in \N: \exists \delta \in \Q_{>0}: \forall h \in \Q \set{0}: \size{h}<\delta \implies \size{\frac{\map f {x+h}-\map f x}{h} -L} < 2^{-n}$.

Let $\set{\delta_n}$ be an enumeration of all such $\delta$s considered.

We claim that the existence of the derivative at a point is implied by:


 * $\forall n \in \N: \exists \delta_n \in \Q_{>0}: \exists y_n \in \Q: \forall h \in \closedint{-\delta_n}{\delta_n}: \size{ \map{f}{x+h}-\map{f}{x}- h y_n } \le \size{h} 2^{-n}$

for some rational sequence $\sequence{y_n}_{n \mathop \in \N}$.