In this paper we study the controllability of the following linear time-varying control system x' = A(t)x + Bu(t), where A(t) = a(t)A and a(.) is a scalar function which is in general not continuous and satisfies the only condition: integral(0)(T) a(s),ds not equal 0, T > 0. A and B are constant matrices of dimension n x n and n x m respectively. Under this condition, we prove that this system is controllable on [0, T] iff the usual Rank condition holds: Rank[B 3 vertical dots AB 3 vertical dots...3 vertical dots A(n-1)B] = n.