# continuous derivative implies bounded variation

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

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

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 ContinuousDerivativeImpliesBoundedVariation 2013-03-22 17:56:32 2013-03-22 17:56:32 pahio (2872) pahio (2872) 8 pahio (2872) Theorem msc 26A45 ProductAndQuotientOfFunctionsSum