Question

If f is continuous on [a,b] and f' is bounded in(a,b) prove that f is of bounded variation on [a,b]

Answer 1

Since $f'$ is bounded,

$\exist \, M>0 \ni | f'(\alpha)| \leq M \, \forall \alpha \in [a,b]$

Let $P = \{ x_0,x_1,...,x_n \}$ be a partition of $[a,b]$

Then by the Mean Value Theorem, we choose $\alpha \in (x_{i-1},x_i) \, \ni$ $f(x_i) - f(x_{i-1}) = f'(\alpha)(x_i - x_{i-1})$

Therefore$|f(x_i) - f(x_{i-1})| = |f'(\alpha)||x_i - x_{i-1}| \leq M|x_i - x_{i-1}|$

Hence, we have

$\sum_{i=1}^{n}|f(x_i) - f(x_{i-1})| \leq M \sum_{i=1}^{n} |x_i - x_{i-1}| = M(b-a)$

$V(f,a,b,P) \leq M(b-a)$

Since $M(b-a) > 0$ we have that f is of bounded variation.