Let $f:[0,1] \rightarrow \mathbb{R}$ be a Riemann integrable function and let $\epsilon >0$. Show that there exist continuous functions $g,h:[0,1] \rightarrow \mathbb{R}$ such that $g(x)\leq f(x) \leq h(x)$ for all $x \in [0,1]$, and $\int_{0}^{1} (h(x)-g(x))dx < \epsilon$.
Since $f$ is a Riemann integrable. For all $\epsilon >0$, there exist a partition $P_{\epsilon}$ of $[0,1]$ such that $U(P_{\epsilon},f)-L(P_{\epsilon},f)<\epsilon$, and $L(P_{\epsilon},f)\leq \int_{0}^{1}fdx\leq U(P_{\epsilon},f)$.
Now I want to show that there exist $g,h$ are continuous, I am confused here, how I claim my $g(x)=U(P_{\epsilon},f)$. and it is continuous.
Can anyone suggest me some hint for existence of two continuous function here.
$L(P_\epsilon,f)$ is the integral of a step function $G\le f$ that is constant on each of the intervals of the partition $P_\epsilon$. To get $g$, modify $G$ near each of its jumps to make it continuous. Thus if $G(x) = d$ for $a \le x \le b$ and $e$ for $b < x \le c$, with $d < e$, take $g(x) = d + m (x-b)$ for $b \le x \le b + (e-d)/m$ for suitable $m$. Similarly on the other side to get $h$.