continuous derivative implies bounded variation
Theorem. If the real function $f$ has continuous^{} derivative on the interval $[a,b]$, then on this interval,

•
$f$ is of bounded variation^{},

•
$f$ can be expressed as difference of two continuously differentiable monotonic functions.
Proof. ${1}^{\underset{\xaf}{o}}$. The continuous function ${f}^{\prime}$ has its greatest value $M$ on the closed interval^{} $[a,b]$, i.e.
$${f}^{\prime}(x)\leqq M\mathit{\hspace{1em}}\forall x\in [a,b].$$ 
Let $D$ be an arbitrary partition of $[a,b]$, with the points
$$ 
Consider $f$ on a subinterval $[{x}_{i1},{x}_{i}]$. By the meanvalue theorem, there exists on this subinterval a point ${\xi}_{i}$ such that $f({x}_{i})f({x}_{i1})={f}^{\prime}({\xi}_{i})({x}_{i}{x}_{i1})$. Then we get
$${S}_{D}:=\sum _{i=1}^{n}f({x}_{i})f({x}_{i1})=\sum _{i=1}^{n}{f}^{\prime}({\xi}_{i})({x}_{i}{x}_{i1})\leqq M\sum _{i=1}^{n}({x}_{i}{x}_{i1})=M(ba).$$ 
Thus the total variation satisfies
$$ 
whence $f$ is of bounded variation on the interval $[a,b]$.
${2}^{\underset{\xaf}{o}}$. Define the functions^{} $G$ and $H$ by setting
$$G:=\frac{{f}^{\prime}+{f}^{\prime}}{2},H:=\frac{{f}^{\prime}{f}^{\prime}}{2}.$$ 
We see that these are nonnegative and that ${f}^{\prime}=GH$. Define then the functions $g$ and $h$ on $[a,b]$ by
$$g(x):=f(a)+{\int}_{a}^{x}G(t)\mathit{d}t,h(x):={\int}_{a}^{x}H(t)\mathit{d}t.$$ 
Because $G$ and $H$ are nonnegative, the functions $g$ and $h$ are monotonically nondecreasing. We have also
$$(gh)(x)=f(a)+{\int}_{a}^{x}(G(t)H(t))\mathit{d}t=f(a)+{\int}_{a}^{x}{f}^{\prime}(t)\mathit{d}t=f(x),$$ 
whence $f=gh$. Since $G$ and $H$ are by their definitions continuous, the monotonic functions $g$ and $h$ have continuous derivatives ${g}^{\prime}=G$, ${h}^{\prime}=H$. 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.
Title  continuous derivative implies bounded variation 

Canonical name  ContinuousDerivativeImpliesBoundedVariation 
Date of creation  20130322 17:56:32 
Last modified on  20130322 17:56:32 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  8 
Author  pahio (2872) 
Entry type  Theorem 
Classification  msc 26A45 
Related topic  ProductAndQuotientOfFunctionsSum 