Limits of the chosen Continuity model
A numerical pass cannot exclude a specially constructed hole or oscillation below the sample scale.
Preserving the Continuity setup
Continuity at a requires f(a) to exist and equal the common two-sided limit. Numerical checks compare these three quantities.
Start the Continuity review with Function f(x). Compare Function f(x) with its source, then test Test point in a second Continuity run without changing the first Continuity case.
Applications of Continuity
Continuity matters when applying intermediate-value arguments, optimization results, and numerical integration methods. A related application of Continuity Checker is limit estimate.
Testing the assumptions behind Continuity
The entries defining Continuity are Function f(x) and Test point. Vary one numerical setting while leaving the mathematical problem unchanged. A defensible result should settle as resolution improves. Treat agreement after rounding alone cautiously when the unrounded estimates continue to move.
For this continuity result, a useful record includes the function, relevant bounds, and the numerical convention used here. Those details matter more than the calculator interface and prevent an approximation from being quoted as a symbolic identity.
Testing Continuity beyond the example
Inspect the domain of Function f(x) before using Continuity. Keep exact Continuity work separate from Test point.
Test Continuity on a constant or linear function. Refine any numerical Continuity step and compare the approximation.
The Continuity case starts with Function f(x) and Test point. Recalculate Continuity assessment from those entries. A nearby Test point can challenge the Continuity relationship, but its Continuity assessment belongs to a separate Continuity record.
Cross-checking Continuity
Use a familiar benchmark for Function f(x) to challenge the Continuity output. Apply the same Test point and anticipate Continuity assessment before calculating. This benchmark need not duplicate the problem; it only needs to reveal an implausible Continuity assessment or reversed Continuity direction.
Check the direction of Continuity assessment by changing Function f(x) slightly. Hold Test point steady during this Continuity trial. The new Continuity assessment should move as the Continuity relationship predicts unless the calculation crosses a stated boundary.