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[parent] continuous derivative implies bounded variation (Theorem)

Theorem. If the real function $ f$ has continuous derivative on the interval $ [a,\,b]$, then on this interval,

Proof. $ 1^{\underline{o}}$. The continuous function $ \vert f'\vert$ has its greatest value $ M$ on the closed interval $ [a,\,b]$, i.e.

$\displaystyle \vert f'(x)\vert \leqq M \quad \forall x \in [a,\,b].$
Let $ D$ be an arbitrary partition of $ [a,\,b]$, with the points
$\displaystyle x_0 = a < x_1 < x_2 < \ldots < x_{n-1} < b = x_n.$
Consider $ f$ on a subinterval $ [x_{i-1},\,x_i]$. By the mean-value theorem, there exists on this subinterval a point $ \xi_i$ such that $ f(x_i)-f(x_{i-1}) = f'(\xi_i)(x_i-x_{i-1})$. Then we get
$\displaystyle S_D := \sum_{i=1}^n\vert f(x_i)-f(x_{i-1})\vert = \sum_{i=1}^n\vert f'(\xi_i)\vert(x_i-x_{i-1}) \leqq M\sum_{i=1}^n(x_i-x_{i-1}) = M(b-a).$
Thus the total variation satisfies
$\displaystyle \sup_{D}\{$all $\displaystyle S_D$'s$\displaystyle \} \leqq M(b-a) < \infty,$
whence $ f$ is of bounded variation on the interval $ [a,\,b]$.

$ 2^{\underline{o}}$. Define the functions $ G$ and $ H$ by setting

$\displaystyle G := \frac{\vert f'\vert+f'}{2}, \quad H := \frac{\vert f'\vert-f'}{2}.$
We see that these are non-negative and that $ f' = G-H$. Define then the functions $ g$ and $ h$ on $ [a,\,b]$ by
$\displaystyle g(x) := f(a)+\int_a^xG(t)\,dt, \quad h(x) := \int_a^xH(t)\,dt.$
Because $ G$ and $ H$ are non-negative, the functions $ g$ and $ h$ are monotonically nondecreasing. We have also
$\displaystyle (g-h)(x) = f(a)\!+\!\int_a^x(G(t)\!-\!H(t))\,dt = f(a)\!+\!\int_a^xf'(t)\,dt = f(x),$
whence $ f = g-h$. Since $ G$ and $ H$ are by their definitions continuous, the monotonic functions $ g$ and $ h$ have continuous derivatives $ g' = G$, $ h' = H$. So $ g$ and $ h$ fulfil the requirements of the theorem.

Remark. It may be proved that each function of bounded variation is difference of two bounded monotonically increasing functions.



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Cross-references: monotonically increasing, bounded, function of bounded variation, definitions, monotonically nondecreasing, functions, total variation, mean-value theorem, subinterval, points, partition, closed interval, monotonic functions, continuously differentiable, difference, bounded variation, interval, derivative, continuous, real function
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This is version 5 of continuous derivative implies bounded variation, born on 2008-03-24, modified 2008-03-25.
Object id is 10439, canonical name is ContinuousDerivativeImpliesBoundedVariation.
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Classification:
AMS MSC: 26A45 (Real functions :: Functions of one variable :: Functions of bounded variation, generalizations)
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