hey
i would say:
the 2nd $F(x)$ is just a way you can define those $F(x)$, so
$F(x) = \Theta(x)$, if $\Theta$ is continues, say at the Intervals $I_i$, $i\in\mathbb R$
on the intervals $I_i$ you connect the ends on the lefthandside and on the righthandside so that is linear, so for...