Define a function $g : [0, 1] \mapsto \mathbb R$ by the following formula:
$$g(x)=\begin{cases}-1,&x\in \mathbb{Q}\\x^3-x,&x\not\in \mathbb{Q}\end{cases}$$
(a) What is $\underline{I}^b_a(g)$ ? Justify your answer from the definition.
(b) What is $\overline{I}^b_a(g)$ ? Justify your answer from the definition.
(c) Is $g$ integrable on $[0, 1]$? Justify your answer
What I tried so far:
Let $p$ be any partition
Let $p_n=\{x_0,...,x_n\}$ (equal sub-intervals)
s.t. for each index $i_1,i_2 \in [1,n]\cap \mathbb{N}$ have $\triangle x_{i_1}=\triangle x_{i_2}$
(a)
\begin{align} \underline{I}^b_a(g) & =\sup \{\text{lower sum of }g(x)\} \\ & =\lim _{\|p\|\to 0}L_p(g)\tag{*} \\ & =\lim _{n\to \infty }\underline{S^*_{p_{n}}}(g) \Leftrightarrow =\lim _{n\to \infty }L_{p_n}(g) \tag{**} \\ & =\lim _{n\to \infty }\sum _{i=1}^n\:\frac{b-a}{n}\inf\limits _{x\in[x_{i-1},x_i]}g(x) \tag{***} \\ & =\lim _{n\to \infty }\sum _{i=1}^n\:\frac{b-a}{n}(-1) \\ & =a-b \end{align}
(b)
\begin{align} \overline{I}^b_a(g) & = \inf \{\text{upper sum of }g(x)\} \\ & =\lim _{\|p\|\to 0}U_p(g) \tag{*} \\ & =\lim _{n\to \infty }\overline{S^*_{p_{n}}}(g) \Leftrightarrow =\lim _{n\to \infty }U_{p_n}(g) \tag{**} \\ & =\lim _{n\to \infty }\sum _{i=1}^n\:\frac{b-a}{n}\sup\limits_{x\in[x_{i-1},x_i]}g(x) \tag{***} \\ & =\lim _{n\to \infty }\sum _{i=1}^n\:\frac{b-a}{n}\sup\limits_{x\in[x_{i-1},x_i]}x^3-x \\ & =\overline{I}^b_a(x^3-x)=\underline{I}^b_a(x^3-x) \\ & = {I}^b_a(x^3-x) \\ & =\lim _{n\to \infty }\sum _{i=1}^n\:\frac{b-a}{n}(x^3_i-x_i) \\ & \vdots \\ & =-\frac{a^4}{4}+\frac{b^4}{4}+\frac{a^2}{2}-\frac{b^2}{2} \end{align}
(c)No, since lower integral and upper integral are not equal.
Question:
1.From $(*)$ to $(**)$, is it right to say that when $\|P\|$ approaches $0$ for lower Darboux sum that is same as the lower Riemann sum when n approaches infinite? And the same for upper Darboux sums?
2.From $(**)$ to $(***)$, did I write the right definitions?
3.Is there any other metheds?
Any help would be appreciated.
Update1
For my first question, I think Upper/Lower Riemann sum is same as Upper/Lower Darboux sum, that notation $\overline{S^*_{p}}(g)$/$\underline{S^*_{p}}(g)$ means same thing as $U_p(g)$/$L_p(g)$, but is it valid to change $\|P\|\rightarrow 0$ to $n \rightarrow \infty$ and $p$ to $p_n$? For my second question, the definition should be right.
Update2
Step in Q1 and Q2 are both valid, I just wondering if there is other methed to do this.