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

|f^{\prime}(x)|\leqq M\quad\forall x\in[a,\,b].

Let $D$ be an arbitrary partition of $[a,\,b]$ , with the points

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 $f(x_{i})-f(x_{i-1})=f^{\prime}(\xi_{i})(x_{i}-x_{i-1})$ . Then we get

S_{D}:=\sum_{i=1}^{n}|f(x_{i})-f(x_{i-1})|=\sum_{i=1}^{n}|f^{\prime}(\xi_{i})|% (x_{i}-x_{i-1})\leqq M\sum_{i=1}^{n}(x_{i}-x_{i-1})=M(b-a).

Thus the total variation satisfies

\sup_{D}\{\mbox{all }S_{D}\mbox{'s}\}\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

G:=\frac{|f^{\prime}|+f^{\prime}}{2},\quad H:=\frac{|f^{\prime}|-f^{\prime}}{2}.

We see that these are non-negative and that $f^{\prime}=G-H$ . Define then the functions $g$ and $h$ on $[a,\,b]$ by

g(x):=f(a)+\int_{a}^{x}G(t)\,dt,\quad h(x):=\int_{a}^{x}H(t)\,dt.

Because $G$ and $H$ are non-negative, the functions $g$ and $h$ are monotonically nondecreasing. We have also

(g-h)(x)=f(a)\!+\!\int_{a}^{x}(G(t)\!-\!H(t))\,dt=f(a)\!+\!\int_{a}^{x}f^{% \prime}(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 $g^{\prime}=G$ , $h^{\prime}=H$ . 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	2013-03-22 17:56:32
Last modified on	2013-03-22 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