Vtgyjfy

Let $f:[a,b]to mathbb{R}$ be a monotone function (say strictly increasing).

Then, do for every $epsilon>0$ exist two step functions $h,g$ so that $gle fle h$ and $0le h-gle epsilon$?

Does there exist some closed form of these functions like the one below?

I encountered the problem when trying to prove the Riemann Integrability of monotone functions via the traditional definition of the Riemann Integral (not the one with Darboux sums). The book I am reading proves that a continuous function is integrable via the process below:

Let $mathcal{P}=left{ a=x_0<...<x_n=b right}$ be a partition of $[a,b]$. Define $m_i=min_{xin [x_{i-1},x_i]}f(x)$ and $M_i=max_{xin [x_{i-1},x_i]}f(x)$. By the Extreme Value theorem $M_i,m_i$ are well defined. We approximate $f$ with step functions:
begin{gather}g=m_1chi_{[x_0,x_1]}+sum_{i=2}^nm_ichi_{(x_{i-1},x_i]}\
h=M_1chi_{[x_0,x_1]}+sum_{i=2}^nM_ichi_{(x_{i-1},x_i]}end{gather}
It is easily seen that they satisfy the "step function approximation" and as $0le int_a^bh-gle epsilon$ (uniform continuity) by a previous theorem $f$ is integrable.

The book then goes on to generalise by discussing regulated functions. I would like however to see a self contained proof similar to the previous one if $f$ is monotone. This question is reduced to the two questions asked in the beggining of the post

Let $varepsilon>0$ and $N_varepsilon$ the smallest $n in mathbb{N}$ such that
$$
frac{1}{n}(b-a)(f(b)-f(a)) le varepsilon
$$
For $n ge max{2,N_varepsilon}$ consider the following partition of $[a,b]$:
$$
mathcal{P}={a=x_0<x_1<ldots<x_n=b}, x_i=a+ifrac{b-a}{n} 0 le i le n.
$$
Set
$$
A_i=begin{cases}
[x_0,x_1] & text{ for } i=0\
(x_i,x_{i+1}]& text{ for } 1 le i le n-1
end{cases}
$$
Since $f$ is strictly increasing for each $i in {0,ldots,n-1}$ we have
$$
f(x_i) le f(x) le f(x_{i+1}) quad forall x_i le x le x_{i+1}.
$$
Setting
$$
h=sum_{i=0}^{n-1}f(x_{i+1})chi_{A_i}, g=sum_{i=0}^{n-1}f(x_i)chi_{A_i}
$$
we have
$$
g(x)le f(x) le h(x) quad forall x in [a,b],
$$
and
$$
h(x)-g(x)=sum_{i=0}^{n-1}Big(f(x_{i+1})-f(x_i)Big)chi_{A_i}(x)>0 quad forall x in [a,b].
$$
In addition
begin{eqnarray}
int_a^b(h-g)&=&sum_{i=0}^{n-1}(f(x_{i+1})-f(x_i))int_a^bchi_{A_i}=sum_{i=0}^{n-1}(f(x_{i+1})-f(x_i))(x_{i+1}-x_i)\
&=&frac{b-a}{n}sum_{i=0}^{n-1}(f(x_{i+1})-f(x_i))=frac{1}{n}(b-a)(f(b)-f(a))\
&le&frac{1}{N_varepsilon}(b-a)(f(b)-f(a))le varepsilon.
end{eqnarray}

Let $P$ be a finite partition of $[f(a),f(b)]$ into subintervals of length at most $epsilon$. For $Ain P$, write $m_A<M_A$ for the endpoints of $A$ and define $$g=sum_{Ain P}m_Achi_{f^{-1}(A)}qquadmbox{and}qquad h=sum_{Ain P} M_Achi_{f^{-1}(A)}.$$