I am solving previous year questions from ISI JRF exam.One of the question is:
Let $f:[a,b]\to \mathbb R$ be continuous and let $r_1,r_2,...,r_n\in f[a,b]$ then show that $f(x_0)=\frac{r_1+r_2+...+r_n}{n}$ for some $x_0\in [a,b]$.
I have solved the problem but it is so easily solved that I doubt whether my answer is correct.I have solved this as follows:
Order these $n$ numbers say $r_1\leq r_2\leq...\leq r_n$ then the number $\frac{r_1+r_2+...+r_n}{n}$ is the mean of these $n$ numbers and so it lies between $r_1$ and $r_n$.But $f$ attains both $r_1$ and $r_n$ so $f$ must also attain any value in between by intermediate value property of continuous functions.
Is this solution ok?