What kind of functions can be Riemann integrable? I have learned that every continuous, or piecewise continuous function can be Riemann integrated.
But then, are there uncontinuous functions that are Riemann integrable? And if there is, can I still understand the value of a definite Riemann integral as 'an area under a curve'? I'm not sure I can imagine a totally uncontinuous function having 'area under its function' in my head
Thanks in advance
 A: A function $f$ is Riemann integrable on the interval $[a, b]$ if the following condition holds: For any partition $P = \{ x_{0}, ..., x_{n} \}$ of $[a, b]$, we have an $\epsilon > 0$ such that $U(P, f) - L(P, f) < \epsilon$.
Here, $U(P, f) = \sum_{i=1}^{n} M_{i} \Delta x_{i}$, for $M_{i} = sup \{ f(x) : x \in [x_{i-1}, x_{i}]$. The $sup$ is the supremum, or least upper bound.
Similarly, $L(P, f) = \sum_{i=1}^{n} M_{i} \Delta x_{i}$ for $m_{i} = inf \{ f(x) : x \in [x_{i-1}, x_{i}] \}$. The $inf$ is the infimum, or greatest lower bound. 
Conceptually, we are just taking Riemann sums. $U(P, f)$ is a Riemann sum where, for each given interval, we take the largest value. Similarly, $L(P, f)$ is a Riemann sum where we take the smallest value on each interval. So essentially, if we can control how much these two Riemann sums differ, we can integrate $f$.
As for conceptualizing this, I'd think about Riemann integrals in this way: you add up the rate and you get a change. That's really what Calculus is about. 
A: *

*continuous function on closed interval

*monotonic and bounded function on closed interval.

*bounded functions that satisfy 1. or 2. on every sub-interval that constitutes [a,b].
4.others.
