Differentiable Function with Bounded Derivative is of Bounded Variation
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Theorem
Let $a, b$ be real numbers with $a < b$.
Let $f : \closedint a b \to \R$ be a continuous function.
Let $f$ be differentiable on $\openint a b$, with bounded derivative.
Then $f$ is of bounded variation.
Proof
For each finite subdivision $P$ of $\closedint a b$, write:
- $P = \set {x_0, x_1, \ldots, x_n }$
with:
- $a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$
Since the derivative of $f$ is bounded, there exists some $M \in \R$ such that:
- $\size {\map {f'} x} \le M$
for all $x \in \openint a b$.
By the Mean Value Theorem, for each $i \in \N$ with $i \le n$, there exists $\xi_i \in \openint {x_{i - 1} } {x_i}$ such that:
- $\map {f'} {\xi_i} = \dfrac {\map f {x_i} - \map f {x_{i - 1} } } {x_i - x_{i - 1} }$
Note that, from the boundedness of $f'$:
- $\size {\map {f'} {\xi_i} } \le M$
We also have from the fact that $x_i > x_{i - 1}$:
- $\size {x_i - x_{i - 1} } = x_i - x_{i - 1}$
So, for each $i$:
\(\ds \size {\map f {x_i} - \map f {x_{i - 1} } }\) | \(=\) | \(\ds \size {\map {f'} {\xi_i} } \paren {x_i - x_{i - 1} }\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds M \paren {x_i - x_{i - 1} }\) |
We therefore have:
\(\ds \map {V_f} {P ; \closedint a b}\) | \(=\) | \(\ds \sum_{i \mathop = 1}^n \size {\map f {x_i} - \map f {x_{i - 1} } }\) | using the notation from the definition of bounded variation | |||||||||||
\(\ds \) | \(\le\) | \(\ds M \sum_{i \mathop = 1}^n \paren {x_i - x_{i - 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds M \paren {x_n - x_0}\) | Telescoping Series: Example 2 | |||||||||||
\(\ds \) | \(=\) | \(\ds M \paren {b - a}\) |
for all finite subdivisions $P$.
So $f$ is of bounded variation.
$\blacksquare$
Also see
- Differentiable Function of Bounded Variation may not have Bounded Derivative: demonstrating that while this theorem gives a simple sufficient condition for a differentiable continuous function $f$ to be of bounded variation, it is not a necessary condition.
Sources
- 1973: Tom M. Apostol: Mathematical Analysis (2nd ed.) ... (previous) ... (next): $\S 6.3$: Functions of Bounded Variation: Theorem $6.6$