Show that there exists a partition and sample points such that the Riemann sum equals any intermediate value

Question

Let $f:[0, 1]\rightarrow \mathbb{R}$ be a bounded function. Show that for any c such that $f(a) < c < f(b)$ where $a,b \in [0, 1]$ there exists a partition $P$ and sample points such that the Riemann sum is equal to c.

I tried proving this by making 2 or 3 subintervals containing $a$, $b$, and an endpoint such that the Riemann sum will be c but I cannot seem to figure it out.

Answer 1

$c$ may be written as $c=tf(a)+(1-t)f(b)$ with $0<t<1$. This is a Riemann sum with partition $P=\{0 ,t, 1\}$ and sample points $a,b$ if $a\leq t\leq b$. (This is only a begining.)