Riemann condition and integration

Question

$$ f(x) = \begin{cases} x^2, & 0 ≤ x ≤ 1 \\[2ex] x+1, & 1 < x ≤ 2 \end{cases} $$

Let $ε > 0$ be given

Question

Let $n ∈ \mathbb N$ be given. Explain why there is a Partition $P = \{\mathbf{X}_0,\mathbf{X}_1,\mathbf{X}_2,\ldots,\mathbf{X}_k\}$ of the interval $[0, 1- 1/n]$ for which $\sum_{i=1}^k (M_i-m_i)(X_i - X_{i-1}) < 1/3ε$

how do i even start with this ? Sorry for reposting I made a error in the previous post so the question did not make sense

I think the following theorem may be of use:

Let $f$ be defined and bounded on $[a,b]$. Then $f$ is Riemann integrable on $[a,b]$ iff for every $ε > 0$ there exists a partition $P$ of $[a,b]$ such that $U(P) - L(P) < ε$

Answer 1

For fixed $n \in \mathbb{N}$, the function $f$ coincides with $x \mapsto x^2$ on the interval $[0,1-1/n]$. If you know that a continuous function is Riemann integrable on a closed interval, you are done. Otherwise, you could also try to mimick the proof that an increasing function is integrable, as you can read in Chapter 6 of Rudin's Principles of Mathematical Analysis.