Let $f(x)$ be a continuous increasing function on $ \left[a, \;b \right]$, it is obvious that $ f(a)(b-a) < \int_a ^b f(x)dx < f(b)(b-a) $.
On the other hand, I want to show that:
Let $ p, \;q $ be constants ($p < q$), for all $ k $ that satisfies $ p(b-a) < k < q(b-a)$, there exists continuous increasing function $ f $ such that $ f(a) = p, \;f(b) = q,\; \int_{a}^{b} f(x)dx = k$
It seems very natural when I sketch some graphs of f, but I cannot prove this. How can prove or disprove this statement?