We can use the following:
Lemma. If $f$ is continuous on $[a,b]$ then for any $\epsilon > 0$ there exists $\delta > 0$ such that for any partition $P$ of $[a,b]$ with $\|P\| < \delta$ and any refinement $R$ of $P$, we have $|S(f,P) - S(f,R)| < \epsilon$.
Applying the lemma, we can show that if $f$ is continuous, then the Cauchy criterion is satisfied. That is, for any $\epsilon >0$ there exists $\delta > 0$ such that if $P$ and $Q$ are any partitions satisfying $\|P\|, \|Q\| < \delta$, then $|S(f,P) - S(f,Q)| < \epsilon.$
To see this, let $R = P \cup Q$ be a common refinement and take $\delta$ as specified in the lemma such that if $\|P\|, \|Q\| < \delta$, we have $|S(f,P) - S(f,R)| < \epsilon/2$ and $|S(f,Q) - S(f,R)| < \epsilon/2$. Whence, it follows that
$$|S(f,P) - S(f,Q)| \leqslant |S(f,P) - S(f,R)| + |S(f,Q) - S(f,R)| < \epsilon/2 + \epsilon/2 = \epsilon.$$
It remains to prove the lemma.
Since $[a,b]$ is compact, $f$ is uniformly continuous and for any $\epsilon > 0$ there exists $\delta >0$ such that if $|x - y| < \delta$, then $|f(x) - f(y)| < \epsilon/(b-a)$. Suppose $\|P\| < \delta$ and $R$ is a refinement of $P$. Any subinterval $[x_{j-1}, x_{j}]$ of $P$ can be decomposed as the union of subintervals of $R$,
$$[x_{j-1},x_{j}] = \bigcup_{k=1}^{n_j}[y_{j,k-1}, y_{j,k}],$$
and
$$\begin{align}\left|S(f,P) - S(f,R)\right| &= \left|\sum_{j=1}^n f(\xi_j)(x_j - x_{j-1}) - \sum_{j=1}^n \sum_{k=1}^{n_j}f(\eta_{j,k})(y_{j,k} - y_{j,k-1})\right| \\ &\leqslant \sum_{j=1}^n \sum_{k=1}^{n_j}|f(\xi_j) - f(\eta_{j,k})|(y_{j,k} - y_{j,k-1}) \\ &\leqslant \sum_{j=1}^n \sum_{k=1}^{n_j}\frac{\epsilon}{b-a}(y_{j,k} - y_{j,k-1}) \\ &= \epsilon\end{align}$$
Best Answer
A function $f:[a,b]\to \mathbb{R}$ is Riemann integrable if for any $\varepsilon >0$ there exists a partition $\mathcal{P}$ of $[a,b]$ such that $$U_{[a,b]}(\mathcal{P},f)-L_{[a,b]}(\mathcal{P},f)<\varepsilon$$ where $U$ and $L$ denote the upper and the lower Riemann sums.
Let $M=\sup |f(x)|.$ For fixed $\varepsilon >0$ let $c$ be chosen so that $c\le a+{\varepsilon\over 4M}.$ By assumptions there is a partition $\mathcal{P}_0$ of $[c,b]$ such that $$U_{[c,b]}(\mathcal{P}_0,f)-L_{[c,b]}(\mathcal{P}_0,f)<{\varepsilon\over 2}$$ Then for $\mathcal{P}=\{a\}\cup \mathcal{P}_0$ we have $$U_{[a,b]}(\mathcal{P},f)-L_{[a,b]}(\mathcal{P},f)\\ \le {\varepsilon\over 2}+U_{[c,b]}(\mathcal{P}_0,f)-L_{[c,b]}(\mathcal{P}_0,f)<\varepsilon $$