In proof of (change of variable in integration, which i uploaded ) which definition they used in equation (3) to show that function on R.H.S in equation (3) is integrable?
My thoughts : if we already know the function we have is Reimann integrable, then to find its integral, we can restrict whatever we want (just it has to be subnet).and we can find its integral which is done in L.H.S Of equation(3).
(1) Now what i mean by restriction: by restriction i mean we can find limit using only specific partitions(like equal sub interval).Or we can choose tags in some specific way(and partitions as well ). I know if function we have is integrable then this all are just a subnet, so they will bring me to correct answer.
But in equation (3) they claimed function on R.H.S is integrable ,by using specific tags(as you can see tags depend on Mean value theorem). So My question is How this is sufficient to show that function is Reimann integrable?
The proof in your book has a subtle flaw as it does not assume the arbitrary tags $\eta_i$. One approach would be to assume the integrability of $f(\phi(y)) \phi'(y) $ and proceed as in the book.
Here is a way to deal with the situation in proper manner without assuming any extra hypotheses. Let's keep the notation as in your textbook and choose a random partition $Q=\{y_0,y_1,\dots,y_n\}$ of $[\alpha, \beta] $ and let $P=\{x_0,x_1,\dots,x_n\}$ be the corresponding partition of $[a, b] $ so that $x_i=\phi(y_i) $. Note further that $$\Delta x_i=x_i-x_{i-1}=\phi(y_i)-\phi(y_{i-1})=\phi'(\theta_i)\Delta y_i\tag{1}$$ for some $\theta_i\in[y_{i-1},y_i]$. Since $\phi'$ is Riemann integrable over $[\alpha, \beta] $ therefore it is also bounded over same interval and let $|\phi'(y) |< K$ for all $y\in[\alpha, \beta] $. We then have $$\Delta x_i< K\Delta y_i\tag{2}$$ Let $\eta_i\in [y_{i-1},y_i]$ be arbitrary tag points corresponding to partition $Q$ of $[\alpha, \beta] $ and let $\xi_i=\phi(\eta_i) $. Let $\epsilon >0$ be any pre-assigned arbitrary number. Then by Riemann integrability of $f$ on $[a, b] $ there exists a corresponding $\delta_1>0$ such that $$\left|\sum_{i=1}^{n}f(\xi_i)\Delta x_i-\int_a^bf(x)\,dx\right|<\frac{\epsilon} {2}\tag{3}$$ whenever $\mu(P) <\delta_1$.
Let $M$ be a positive upper bound for $|f|$ on $[a, b] $. Let $U(\phi', Q), L(\phi', Q) $ denote the upper and lower Darboux sums for $\phi'$ over partition $Q$ of $[\alpha, \beta] $ so that $$U(\phi', Q) =\sum_{i=1}^{n}M_i\Delta y_i, L(\phi', Q) =\sum_{i=1}^{n}m_i\Delta y_i$$ where $M_i, m_i$ are supremum and infimum of $\phi'$ on $[y_{i-1},y_i]$. Since $\phi'$ is Riemann integrable on $[\alpha, \beta] $ there exists a $\delta_2>0$ such that $$U(\phi', Q) - L(\phi', Q) <\frac{\epsilon} {2M}\tag{4}$$ whenever $\mu(Q) <\delta_2$.
We now analyze the expression $$S=\left|\sum_{i=1}^n f(\phi(\eta_i)) \phi'(\eta_i) \Delta y_i-\int_{a} ^{b} f(x) \, dx\right|$$ Let $\delta=\min(\delta_1/K, \delta_2)$ and let $\mu(Q) <\delta$ so that by $(2)$ we have $\mu(P) <\delta_1$ and \begin{align} S&\leq \left|\sum_{i=1}^n f(\phi(\eta_i)) \phi'(\eta_i) \Delta y_i-\sum_{i=1}^n f(\xi_i) \Delta x_i\right|+\left|\sum_{i=1}^n f(\xi_i) \Delta x_i-\int_a^b f(x) \, dx\right|\notag\\ &<\left|\sum_{i=1}^n f(\phi(\eta_i)) ((\phi'(\eta_i) - \phi'(\theta_i)) \Delta y_i\right|+\frac{\epsilon} {2}\text{ (via (1) and (3))}\notag\\ &\leq \sum_{i=1}^n|f(\phi(\eta_i))||\phi'(\eta_i)-\phi'(\theta_i)|\Delta y_i+\frac{\epsilon} {2}\notag\\ &\leq M \sum_{i=1}^n(M_i-m_i)\Delta y_i+\frac{\epsilon} {2}\notag\\ &= M(U(\phi', Q) - L(\phi', Q) +\frac{\epsilon} {2}\notag\\ &<M\cdot\frac{\epsilon} {2M}+\frac{\epsilon} {2}\text{ (via (4))}\notag\\ &=\epsilon\notag \end{align} It now follows that $f(\phi(y)) \phi'(y) $ is Riemann integrable over $[\alpha, \beta] $ with integral $\int_a^b f(x) \, dx$.
The above also proves that if $\phi'$ is bounded away from zero in addition to the given hypotheses then $f(g(y)) $ is Riemann integrable on $[\alpha, \beta] $.