How do I formulate this sentence in a formal mathematical way: "$\delta x$ is the largest band in $x$ that satisfies $f(x) < 1/2$"? That is, how do I define $\delta x$?
Take the following figure as an example, where $x$ goes from 0 to 20. We have $\tilde{\delta}x_1 = 5 -3=2$ and $\tilde{\delta}x_2 = 10 -6=4$. Therefore $\delta x = \max [\tilde{\delta}x_i] = \tilde{\delta}x_2 = 4$.
P.S.: my $f(x)$ is not actually an analytical equation, it is a series of discrete points obtained from a numerical problem.