RMO 2008, Grade 12, Problem 1
Let and be a continuous function on and having Darboux property on . Prove that if and for all nonnegative we have then admits primitives on .
Proof. The title condition is almost entirely redundant- it simply says that .
Obviously, is an antiderivative away from zero. The only interesting question is the one-sided derivative . Now, by hypothesis. We also have by the same integral inequality. Since is increasing (and bounded below), it has a limit at zero. Since the is zero, this limit is zero, and has the proper one-sided derivative at zero.