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)

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
Let $F: \R \times \R \setminus \set 0 \to \R$ be defined as:


 * $\map F {x, h} = \dfrac {\map f {x + h} - \map f x} h$

By the definition of derivative:


 * $\map {f'} x$ exists




 * $\displaystyle \lim_{h \mathop \to 0} \map F {x, h}$ exists.

The second limit can written as:


 * $\exists y \in \R: \forall \varepsilon > 0: \exists \delta > 0: \forall h \in \R: \size h < \delta \implies \size {\map F {x, h} - y} < \epsilon$

Let $\sequence {y_n}: \Q \to \R$ be a Rational Sequence converging to the $y$ whose existence is asserted such.

Such a sequence exists by Rationals are Everywhere Dense in Reals.

[[Category:Continuity [[Category:Sigma-Algebras