Editing Total variation (section)

====The form of the total variation of a differentiable function of one variable====
{{EquationRef|5|Theorem 1.}} The '''total variation''' of a [[differentiable function]] <math>f</math>, defined on an [[interval (mathematics)|interval]] <math> [a , b] \subset \mathbb{R}</math>, has the following expression if <math>f'</math> is Riemann integrable

:<math> V_a^b(f) = \int _a^b |f'(x)|\mathrm{d}x</math>

If <math> f</math> is differentiable and [[Monotonic function|monotonic]], then the above simplifies to
:<math> V_a^b(f) = |f(a) - f(b)|</math>

For any differentiable function <math>f</math>, we can decompose the domain interval <math>[a,b]</math>, into subintervals <math>[a,a_1], [a_1,a_2], \dots, [a_N,b]</math> (with <math>a<a_1<a_2<\cdots<a_N<b </math>) in which <math>f</math> is locally monotonic, then the total variation of <math> f</math> over <math>[a,b]</math> can be written as the sum of local variations on those subintervals:
:<math>
\begin{align}
V_a^b(f) &= V_a^{a_1}(f) + V_{a_1}^{a_2}(f) + \, \cdots \, +V_{a_N}^b(f)\\[0.3em]
&=|f(a)-f(a_1)|+|f(a_1)-f(a_2)|+ \,\cdots \, + |f(a_N)-f(b)|
\end{align}</math>