
riemann integration
I have this question:
If a(x) = x and b(x) = x and if a(x) <= f(x) <= b(x) for all x in [0,1], does it follow that f is riemann integrable for [0,1]?
My textbook has the following squeeze theorem for integrals:
Let f be a function on [a,b]. Then f is riemann integrable on [a,b] if and only if for every epsilon > 0, there exist functions a and b riemann integrable on [a,b] with a(x) <= f(x) <= b(x) for all x in [a,b] and such that
integral from a to b of (b(x)  a(x)) < epsilon.
Since the integral from 0 to 1 of (2x) = 1, which can be made less than epsilon, doesn't this imply the function f is riemann integrable for [0,1]?
Thanks for any help in advance.