In the rectangular coordinate plane, the graphs of functions $f$ and $g$ defined on the closed interval $[0,1]$ and the line $y = x$ are given below.
For real numbers $a, b$ and $c$ in the open interval $(0,1)$
$$\begin{aligned}
& a < f(a) < g(a) \\
& g(b) < b < f(b) \\
& c < g(c) < f(c)
\end{aligned}$$
If these inequalities are satisfied, which of the following orderings is correct?
A) $a < b < c$
B) $a < c < b$
C) $b < a < c$
D) $c < a < b$
E) $c < b < a$