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.

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

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

Thus the total variation satisfies

whence $f$ is of bounded variation on the interval  $[a,\,b]$.

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

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

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

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.