Let $f:[a, b] \rightarrow \mathbb{R}$ be a bounded function defined on the closed, bounded interval $[a, b]$ of $\mathbb{R}$. Suppose that for every $\varepsilon>0$ there is a dissection $\mathcal{D}$ of $[a, b]$ such that $S_{\mathcal{D}}(f)-s_{\mathcal{D}}(f)<\varepsilon$, where $s_{\mathcal{D}}(f)$ and $S_{\mathcal{D}}(f)$ denote the lower and upper Riemann sums of $f$ for the dissection $\mathcal{D}$. Deduce that $f$ is Riemann integrable. [You may assume without proof that $s_{\mathcal{D}}(f) \leqslant S_{\mathcal{D}^{\prime}}(f)$ for all dissections $\mathcal{D}$ and $\mathcal{D}^{\prime}$ of $\left.[a, b] .\right]$

Prove that if $f:[a, b] \rightarrow \mathbb{R}$ is continuous, then $f$ is Riemann integrable.

Let $g:(0,1] \rightarrow \mathbb{R}$ be a bounded continuous function. Show that for any $\lambda \in \mathbb{R}$, the function $f:[0,1] \rightarrow \mathbb{R}$ defined by

$$f(x)= \begin{cases}g(x) & \text { if } 0<x \leqslant 1 \\ \lambda & \text { if } x=0\end{cases}$$

is Riemann integrable.

Let $f:[a, b] \rightarrow \mathbb{R}$ be a differentiable function with one-sided derivatives at the endpoints. Suppose that the derivative $f^{\prime}$ is (bounded and) Riemann integrable. Show that

$$\int_{a}^{b} f^{\prime}(x) d x=f(b)-f(a)$$

[You may use the Mean Value Theorem without proof.]