Sum of Functions of Bounded Variation is of Bounded Variation

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Theorem

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

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

Let $\map {V_f} {\closedint a b}$ and $\map {V_g} {\closedint a b}$ be the total variations of $f$ and $g$ respectively, on $\closedint a b$.


Then $f + g$ is of bounded variation on $\closedint a b$ with:

$\map {V_{f + g} } {\closedint a b} \le \map {V_f} {\closedint a b} + \map {V_g} {\closedint a b}$

where $\map {V_{f + g} } {\closedint a b}$ denotes the total variation of $f + g$ 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:

\(\ds \map {V_{f + g} } {P ; \closedint a b}\) \(=\) \(\ds \sum_{i \mathop = 1}^n \size {\map {\paren {f + g} } {x_i} - \map {\paren {f + g} } {x_{i - 1} } }\) using the notation from the definition of bounded variation
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \size {\paren {\map f {x_i} - \map f {x_{i - 1} } } + \paren {\map g {x_i} - \map g {x_{i - 1} } } }\)
\(\ds \) \(\le\) \(\ds \sum_{i \mathop = 1}^n \size {\map f {x_i} - \map f {x_{i - 1} } } + \sum_{i \mathop = 1}^n \size {\map g {x_i} - \map g {x_{i - 1} } }\) Triangle Inequality
\(\ds \) \(=\) \(\ds \map {V_f} {P ; \closedint a b} + \map {V_g} {P ; \closedint a b}\)

Note that since $f$ and $g$ are of bounded variation, there exists $M, K \in \R$ such that:

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

for all finite subdivisions $P$ of $\closedint a b$.

We therefore have:

$\map {V_{f + g} } {P ; \closedint a b} \le M + K$

for all finite subdivisions $P$.

So $f + g$ is of bounded variation.

Note then that:

\(\ds \map {V_{f + g} } {\closedint a b}\) \(=\) \(\ds \sup_P \paren {\map {V_{f + g} } {P ; \closedint a b} }\) Definition of Total Variation of Real Function on Closed Bounded Interval
\(\ds \) \(\le\) \(\ds \sup_P \paren {\map {V_f} {P ; \closedint a b} } + \sup_P \paren {\map {V_g} {P ; \closedint a b} }\)
\(\ds \) \(=\) \(\ds \map {V_f} {\closedint a b} + \map {V_g} {\closedint a b}\) Definition of Total Variation of Real Function on Closed Bounded Interval

$\blacksquare$


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