Types of discontinuity of Riemann Integrable Functions Is it necessary that a Riemann Integrable Function on $[a, b]$ has left and right hand limits at every point in the interval $[a, b]$?
 A: If you are talking about those nontrivial discontinuity, you can consider the characteristic function of cantor set, which is integrable (since it is discontinuous only on a measure zero set), but its discontinuity are not just jump discontiuity, (since set of jump discontinuity can only be countable) as Cantor set is uncountable.
It is continuous as the complement of cantor set in the unit iterval is open, hence it is $0$ everywhere locally outside cantor set, hence continuous.
A: Another example is $$f(x) = \cases{\sin(1/x) & for $x \in (0,1]$\cr 0 & for $x=0$\cr}$$
which is easily seen to be Riemann integrable on $[0,1]$ (even without using Lebesgue's criterion),
but has no limit as $x \to 0+$.
A: The way Riemann integral is introduced in beginner calculus course, it appears that connuity is a key property for a function to be Riemann integrable. However continuity is only a sufficient and not a necessary condition for a function to be Riemann integrable.
Well, discontinuous functions can also be Riemann integrable. Since the idea of Riemann integral is defined for bounded functions, a Riemann integrable can't have an infinite discontinuity. So that leaves us with two types of discontinuities for Riemann integrable functions :


*

*Simple: Here the left and right hand limit of function exist but they may not be equal or equal to the function value.

*Essential: At least one of the left and right hand limits of function at a point does not exist. In other words the function oscillates finitely in the neighbourhood of point under consideration. 


Also note that it is not the type of discontinuity, but the number/amount of discontinuities which matters in deciding Riemann integrability of a function. 
In particular one can establish easily that a bounded function with a finite number of discontinuities in a closed interval is Riemann integrable on that interval.
An example of a function with essential discontinuity is $f(x) =\sin(1/x^2),f(0)=0$. Since it is discontinuous only at $x=0$ and is bounded it is Riemann integrable on every closed bounded interval. 
Also a function can have infinite number of discontinuities and yet be Riemann integrable. A simple example is $f(x) =(1/x) - \lfloor 1/x\rfloor,f(0)=0$. It is Riemann integrable on $[0,1]$ and is discontinuous at $0,1/2,1/3,\dots, 1/n,\dots$.
