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.
Using the reverse triangle inequality we have

$$\begin{align}\sup_{x \in [x_{j-1},x_j]}|f|(x) - \inf_{x \in [x_{j-1},x_j]}|f|(x) \\&amp;=\sup_{x,y \in [x_{j-1},x_j]}| |f(x)|-|f(y)|| \\ &amp;\leqslant \sup_{x,y \in [x_{j-1},x_j]}| f(x)-f(y)| \\ &amp; = \sup_{x \in [x_{j-1},x_j]}f(x) - \inf_{x \in [x_{j-1},x_j]}f(x).\end{align}$$

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 $\epsilon &gt; 0$ there exists a partition $P$ such that the difference of upper and lower Darboux sums satisfies

$$U(P, |f|) - L(P, |f|) \leqslant U(P,f)-L(P,f) &lt; \epsilon.$$

Hence, $|f|$ is Riemann integrable.
Let $\epsilon &gt;0$ be arbitrary

Let $P=\{a=x_0&lt;x_1&lt;x_2&lt;....&lt;x_n=b\}$ be a partition of $[a,b]$ with $||P||&lt;\delta$

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 $\sum_{i=1}^n\{|M_i^{'}-m_i^{'}|\}\Delta x_i\leq \sum_{i=1}^n|M_i-m_i|\Delta x_i&lt;\epsilon$