Find continuous approximation of $f$.

Question

$f(x)$ is bounded on $[a,b]$ and $f(x) \in R[a,b]$. Then for any $\epsilon > 0$ find $g(x)$ and $h(x)$ which are continuous on $[a,b]$ such that:

(a) $g(x) \le f(x) \le h(x)$, $\forall x \in [a,b] $

(b) $\int_a^b [h(x) - g(x)] ds < \epsilon$.

I thought about looking for piecewise linear functions which are decided up to the superior and inferior of $g$ and $h$ on intervals of a certain partition of $[a,b]$ but since $f(x)$ is not necessary continuous I can hardly go through it.

Answer 1

Hint: Take the upper and lower Darboux sums and approximate those with piecewise linear functions.

Answer 2

Hints:

$1).\ $ Draw a picture of of what you want and use it to do the following:

$2).\ $ There is a partition $P=\{a,x_2,\cdots, x_{n-1},b\}$ such that $U(f,P)-L(f,P)<\epsilon.$

$3).\ $ on each subinterval $[x_i,x_{i-1})$, define a piecewise continuous linear function $g$, by considering the linear segments from $(x_i,f(x_i))$ to $(x_{i-1},f(x_{i-1}))$ for $a=x_1\le x_i\le x_n=b$.

$4).\ $ Show that $L(f,P)\le \int^b_ag(x)dx\le U(P,f)$ so that $\left |\int^b_ag(x)dx-\int^b_af(x)dx \right |<\epsilon$.