Editing Cauchy principal value (section)

===Well-definedness as a distribution===
To prove the existence of the limit 
<math display="block"> \lim_{\varepsilon \to 0^{+}} \int_{\varepsilon}^{+ \infty} \frac{u(x) - u(- x)}{x} \, \mathrm{d}x </math>
for a [[Schwartz function]] <math>u(x)</math>, first observe that <math>\frac{u(x) - u(-x)}{x}</math> is continuous on <math>[0, \infty),</math> as
<math display="block"> \lim_{\,x \searrow 0\,} \; \Bigl[ u(x) - u(-x) \Bigr] ~= ~0 ~</math>
and hence 
<math display="block"> \lim_{x\searrow 0} \, \frac{u(x) - u(-x)}{x} ~=~ \lim_{\,x\searrow 0\,} \, \frac{u'(x) + u'(-x)}{1} ~=~ 2u'(0)~, </math>
since <math>u'(x)</math> is continuous and [[L'Hopital's rule]] applies.

Therefore, <math>\int_0^1 \, \frac{u(x) - u(-x)}{x} \, \mathrm{d}x</math> exists and by applying the [[mean value theorem]] to <math>u(x) - u(-x) ,</math> we get:
:<math> \left|\, \int_0^1\,\frac{u(x) - u(-x)}{x} \,\mathrm{d}x \,\right|
\;\leq\; \int_0^1 \frac{\bigl|u(x)-u(-x)\bigr|}{x} \,\mathrm{d}x
\;\leq\; \int_0^1\,\frac{\,2x\,}{x}\,\sup_{x \in \mathbb{R} }\,\Bigl|u'(x)\Bigr| \,\mathrm{d}x
\;\leq\; 2\,\sup_{x \in \mathbb{R} }\,\Bigl|u'(x)\Bigr| ~. </math>

And furthermore:
:<math> \left| \,\int_1^\infty \frac {\;u(x) - u(-x)\;}{x} \,\mathrm{d}x \,\right| \;\leq\; 2 \,\sup_{x\in\mathbb{R}} \,\Bigl|x\cdot u(x)\Bigr|~\cdot\;\int_1^\infty \frac{\mathrm{d}x}{\,x^2\,} \;=\; 2 \,\sup_{x\in\mathbb{R}}\, \Bigl|x \cdot u(x)\Bigr| ~, </math>

we note that the map
<math display="block"> \operatorname{p.v.}\;\left( \frac{1}{\,x\,} \right) \,:\, {C_{c}^{\infty}}(\mathbb{R}) \to \mathbb{C} </math>
is bounded by the usual seminorms for [[Schwartz functions]] <math> u</math>. Therefore, this map defines, as it is obviously linear, a continuous functional on the [[Schwartz space]] and therefore a [[distribution (mathematics)#Tempered distributions and Fourier transform|tempered distribution]].

Note that the proof needs <math>u</math> merely to be continuously differentiable in a neighbourhood of 0 and <math> x\,u </math> to be bounded towards infinity. The principal value therefore is defined on even weaker assumptions such as <math>u</math> integrable with compact support and differentiable at 0.