Question

Consider the function on [1,5]:

f(x) =
1, x=2
-3, x=4
0, x element of [1,5], x not equal to 2, x not equal to y.

a) Show that for every epsilon greater than 0 there exist a partition P_epsilon of [1,5] such that
L(f, P_epsilon) > -epsilon, and partition Q_epsilon of [1,5] such that U(f, Q_epsilon)< epsilon.

b) Based on part a, prove that f is Reimann integrable on [1,5] and compute the integral.