2. Which of the following conditions are necessary for a function f to be Riemann integrable on the closed interval [a,b], where a < b?

I. f is bounded on [a,b].

II. f is continuous on [a,b].

III. f is differentiable on [a,b].

The Riemann integral of a function f(x) defined on a closed interval [a,b] is given by,

if this limit exist. Where P is a partition of [a,b] whose length is the maximum distance between any two consecutive partition points, and n is the number of partitions.

If f(x) has an infinite discontinuity at some point c in [a,b] then the above limit does not reach a limiting value. However, if f(x) has a finite number of jump discontinuities, then the above limit does reach a limiting value and we say that the function is Riemann integrable.

Thus, the necessary condition for a function f to be Riemann integrable on the closed interval [a,b] is that the function be bounded on [a,b], that is, |f(x)| <= M for all x in [a,b]. So, that the above limit can reach a limiting value regardless of whether the function is continuous or not.