How to solve this Riemann integral problem? Suppose $f$ is Riemann Integrable on $[a,b].$ Is there a real number $c\in [a,b]$ such that 
$$\int_a^c f(x)\,dx=\int_c^b f(x)\,dx$$? If it is, then proof it, otherwise, give a counter example.

If $f$ is Riemann integrable on $[a,b]$ then $f$ is bounded, i.e. there exists an $M$ such that $|f(x)|\leq M$ for all $x\in[a,b]$. This implies that the function
$$F(x):=\int_a^x f(t)\>dt\qquad(a\leq x\leq b)$$
is Lipschitz continuous with constant $M$. Let $\int_a^b f(t)\>dt=:C$. Then $F(a)=0$ and $F(b)=C$. By the intermediate value theorem there is then a $c\in[a,b]$ with $F(c)={C\over2}$ (when $C=0$ choose $c=a$). It is easy to see that we then have
$$\int_a^c f(t)\>dt={C\over2}=\int_c^b f(t)\>dt\ .$$
