If $f$ is Riemann integrable on $[a,b]$ , is $|f|$ Riemann integrable on $[a,b]$? If $f$ is Riemann integrable on $[a,b]$ , is $|f|$ Riemann integrable on $[a,b]$ ?
(The metric is $\mathbb R$ usual)
The other is question is $f$ is Riemann integrable on $[a,b]$ , can I claim $f$ is bounded on $[a,b]$ ? (I think the answer can be either yes or no that depend on considerating generalized function or not)
Update: I think I made a mistake on second question that both definitions of Riemann integral and generalized Riemann integral on [a,b] request the f being bounded on [a,b]. Sorry for my mistake.
 A: Using the reverse triangle inequality we have
$$\begin{align}\sup_{x \in [x_{j-1},x_j]}|f|(x) - \inf_{x \in [x_{j-1},x_j]}|f|(x) \\&=\sup_{x,y \in [x_{j-1},x_j]}| |f(x)|-|f(y)|| \\ &\leqslant \sup_{x,y \in [x_{j-1},x_j]}| f(x)-f(y)| \\ & = \sup_{x \in [x_{j-1},x_j]}f(x) - \inf_{x \in [x_{j-1},x_j]}f(x).\end{align}$$
For a given partition $P = (x_0,x_1, \ldots, x_n)$, the difference of upper and lower sums
is
$$U(P,|f|) - L(P,|f|) = \sum_{j=1}^n \left(\sup_{x \in [x_{j-1},x_j]}|f|(x) - \inf_{x \in [x_{j-1},x_j]}|f|(x)\right)(x_j - x_{j-1})\\ \leqslant  \sum_{j=1}^n \left(\sup_{x \in [x_{j-1},x_j]}f(x) - \inf_{x \in [x_{j-1},x_j]}f(x)\right)(x_j - x_{j-1})\\= U(P,f) - L(P,f)$$
Since $f$ is Riemann integrable, for any $\epsilon > 0$ there exists a partition $P$ such that the difference of upper and lower Darboux sums satisfies
$$U(P, |f|) - L(P, |f|) \leqslant U(P,f)-L(P,f) < \epsilon.$$
Hence, $|f|$ is Riemann integrable.
A: I will use the fact that $f(x)$ is Riemann integrable on $[a,b]$ if and only if it is bounded and continuous almost everywhere on $[a,b]$.
Let $f(x)$ be Riemann integrable on [a,b]. Then $f$ is bounded and continuous almost everywhere. Define $f_+(x)$ by $f_+(x)=f(x)$ if $f(x)>0$, and $f_+(x)=0$ otherwise. Then $f_+(x)$ will also be bounded and continuous almost everywhere. Thus $f_+(x)$ is Riemann integrable. Similarly, define $f_-(x)$ by $f_-(x)=f(x)$ if $f(x)\leq 0$ and $f_-(x)=0$ otherwise. Then $f_-(x)$ is also Riemann integrable by the same reasoning as above. Riemann integrability of $|f(x)|$ follows since $|f(x)|=f_+(x)-f_-(x)$.
A: Let $\epsilon >0$ be arbitrary
Let $P=\{a=x_0<x_1<x_2<....<x_n=b\}$ be a partition of $[a,b]$ with $||P||<\delta$
let $M_i=\sup _{({x_{i-1},x_i})}f$ and $m_i=\inf _{({x_{i-1},x_i})}f$
let $M_i^{'}=\sup _{({x_{i-1},x_i})}|f|$ and $m_i^{'}=\inf _{({x_{i-1},x_i})}|f|$
Then $\sum_{i=1}^n\{|M_i^{'}-m_i^{'}|\}\Delta x_i\leq \sum_{i=1}^n|M_i-m_i|\Delta x_i<\epsilon$
