Multiple of Function of Bounded Variation is of Bounded Variation

Theorem
Let $a, b, k$ be real numbers with $a < b$.

Let $f : \closedint a b \to \R$ be a functions of bounded variation.

Let the total variation of $f$ on $\closedint a b$ be $\map {V_f} {\closedint a b}$.

Then $k f$ is of bounded variation with:


 * $\map {V_{k f} } {\closedint a b} = \size k \map {V_f} {\closedint a b}$

where $\map {V_{k f} } {\closedint a b}$ is the total variation of $k f$ on $\closedint a b$.

Proof
For each finite subdivision $P$ of $\closedint a b$, write:


 * $P = \set {x_0, x_1, \ldots, x_n }$

with:


 * $a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$

Then:

Since $f$ is of bounded variation, there exists $M \in \R$ such that:


 * $\map {V_f} {P ; \closedint a b} \le M$

for all finite subdivisions $P$.

So:


 * $\map {V_{k f} } {P ; \closedint a b} \le \size k M$

So $k f$ is of bounded variation.

We then have: