Let, $y = f(x) = x^2 -2$ be a function and $x$ is related to variable $b$ through the following operation: if $x \ge 0.5$, then $b = 1$ else $b=0$. Thus, the new relation is $g(x) = b = 1$, if $x \ge 0.5$.
I can take the derivative of $y$ w.r.t $x$ as $ x' = dy/dx = 2x$. Can I write
$b' = 2g(x)$ or
$b' = g(x')$ ?
$g(.)$ is a discontinuous function, so I cannot find its derivative.
What is the correct way to express $b'$? I cannot understand how to express $b'$ in terms of $x$