Question

Define the functions of bounded variation. Show that if f is continuous on [a,b], and if 𝑓′ exists and is bounded in the interior, say  𝑓′(𝑥) ≤ 𝐴 for all 𝑥 in  𝑎,𝑏 ,then f is of bounded variation on [a,b].

Answer 1

Solution:

Functions of bounded variation: A function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded: the graph of a function having this property is well behaved in a precise sense.

Theorem: If f is continuous on [a, b] and if $f^{\prime}$ exists and is bounded in the interior, say $\left|f^{\prime}(x)\right| \leq A \forall\ x \in (a, b)$ , then f is of bounded variation on [a, b].

Proof: Applying the Mean-Value Theorem, we have

$\Delta f_{k}=f\left(x_{k}\right)-f\left(x_{k-1}\right)=f^{\prime}\left(t_{k}\right)\left(x_{k}-x_{k-1}\right), \quad \text { where } t_{k} \in\left(x_{k-1}, x_{k}\right) .$

This implies

$\sum_{k=1}^{n}\left|\Delta f_{k}\right|=\sum_{k=1}^{n}\left|f^{\prime}\left(t_{k}\right)\right| \Delta x_{k} \leq A \sum_{k=1}^{n} \Delta x_{k}=A(b-a) .$