1 Riemann Integral
Let \(f:[a,b]\to \mathbb{R}\) be a bounded (not necessarily continuous) function on a compact (closed, bounded) interval. We will define what it means for \(f\) to be Riemann integrable on \([a,b]\) and, in that case, define its Riemann integral \(\int_a^b f\). The integral of \(f\) on \([a,b]\) is a real number whose geometrical interpretation is the signed area under the graph \(y=f(x)\) for \(a\leq x\leq b\).
The Riemann integral is the simplest integral to define and it allows one to integrate every continuous function as well as some not-too-badly discontinuous functions. For highly discontinuous functions however it is insufficient and we instead turn to the Lebesgue integral which is more flexible with better mathematical properties.
1.1 Riemann Integral
We say that two intervals are almost disjoint if they are disjoint or intersect only at a common endpoint.
Let \(I\) be a nonempty compact interval. A partition of \(I\) is a finite collection \(\{I_1, ..., I_n\}\) of almost disjoint, nonempty, compact subintervals whose union is \(I\).
We denote a partition of \([a,b]\) by with subintervals \(I_k=[x_{k-1},x_k]\) either by the collection of intervals \[ \mathcal{P}=\{I_1, ..., I_n\}, \] or by the set of endpoints \[ \mathcal{P}=\{a=x_0 < x_1 < \cdots< x_n = b\}. \] Note that the sum of lengths \(|I_k|=x_k-x_{k-1}\) of the almost disjoint subintervals in a partition of an interval is equal to the length of the whole interval. Suppose that \(f:[a,b]\to\mathbb{R}\) is a bounded function on the compact interval \(I=[a,b]\) with \[ M=\sup_I f\quad \& \quad m=\inf_I f. \] If \(\mathcal{P}\) is a partition of \(I\), let \[ M_k=\sup_{I_k}f\quad \& \quad m_k=\inf_{I_k}f. \] Note that these suprema and infima are well-defined, finite real numbers since \(f\) is bounded. Moreover \[ m\leq m_k\leq M_k<M. \] If \(f\) is continuous on the interval \(I\), then it is bounded and attains its maximum and minimum values on each subinterval, but a bounded discontinuous function need not attain its supremum or infimum.
Further, we define the mesh or net of the partition by \[
||\mathcal{P}||=\max_i(x_i-x_{i-1}).
\] We say that \(f\) is Riemann integrable on \([a,b]\) if there exists a real number \(I\) such that for all sample points \(\xi_i\in[x_{i-1}, x_i]\) \[
\lim_{||\mathcal{P}||\to 0}\sum_{i=1}^n f(\xi_i)(x_i - x_{i-1})=I,
\] that is, as we make the mesh finer and finer this sum (of rectangles) approaches some constant. That common limit (if it exists) is denoted by \[
\int_a^b f(x)~dx.
\] 
## Riemann Integrability
Riemann integrals are the first type of integrals we are introduced to in our study of calculus due to their ease of interpretability and their utility for application to smooth and continuous functions. However, it is not too challenging to define a function that is not Riemann integrable, for example
A bounded function on a compact interval \([a,b]\) is Riemann integrable if and only if it is continuous almost everywhere.
1.2 Properties of Reimann Integrals
The Riemann integral is a linear transformation, that is, if \(f\) and \(g\) are Riemann-integrable on \([a,b]\) and \(\alpha\) and \(\beta\) are constants, then \[ \int_a^b (\alpha f(x)+\beta g(x))~dx = \alpha\int_a^bf(x)~dx + \beta \int_a^b g(x) ~ dx. \]