proof of existence While I was studying for my finals i found the following problem:
Let $f:[a,b]\rightarrow \mathbb{R}$ be a Riemann integrable real function, also let $\displaystyle \int\limits_a^bf(t)\,\mathrm{d}t=3$
Prove that there exist $\displaystyle t_1,t_2 \!\in \! (a,b):\int\limits_{t_1}^{t_2}f(t)\,\mathrm{d}t=1$
Could you give me any hint or a way to start the solution?

To be more careful, let $F(x) = \int_a^xf.$ Then $F$ is continuous on $[a,b]$ with $F(a) = 0, F(b) = 3.$ By the IVT, there exists $t_1\in (a,b)$ such that $F(t_1) = 1/2.$ Because $F$ is continuous on $[t_1,b]$ with $F(t_1) = 1/2, F(b) = 3,$ there is $t_2 \in (x_1,b)$ such that $F(t_2) = 3/2.$ It follows that $1 = F(t_2) - F(t_1) = \int_{t_1}^{t_2} f.$
