Let $f:[a,b]\rightarrow\mathbb{R}$ a function of bounded variation with $\inf_{x\in[a,b]}f(x)>0$. Show that there are two increasing functions $g, h:[a,b]\rightarrow \mathbb{R}$ such that $f=g/h$.
They left me an exercise in my course of real analysis and I don't have idea how to start. Somebody could help me?
Since $C:= \inf_{x \in [a,b]}f(x) > 0$ and $f$ is of bounded variation, the function $$F(x):=\log(f(x))$$ is well-defined and also of bounded variation (proof?). We know that any function of bounded variation can be written in the form $$F(x) = t_1(x) -t_2(x),$$ where $t_1$ and $t_2$ are monotonic increasing function. Thus $$f(x) = \frac{g(x)}{h(x)}$$ with $g(x) = \exp(t_1(x))$ and $h(x)= \exp(t_2(x))$ and both $g$ and $h$ are increasing functions.