A continuous sign change keeps a root inside a nested interval whose length halves every step. The point is to make the formal expression readable enough to audit line by line.
The data
Numerical analysis replaces an exact object \(x^\ast\) by approximations \(x_n\) with controlled error. Stability asks how rounding or data perturbations affect the answer.
There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.
Derivation
The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.
Invariant content
The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.
Scope
Fast local convergence is not a global guarantee. Newton's method can diverge, interpolation can oscillate, and a small residual can coexist with a large forward error.
With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.