Answer to Question #288694 in Real Analysis for Siva

Question

Show that f is continuous on a,b and if f' exists is bounded on interior, says f'x less than A for all x in a,b then the f is of bounded variation

Answer 1

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

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