What question did this study set out to answer?

To provide an exact integral representation for the unique zero of a continuous function on a specific interval.

May 10, 2026Open Access

An Integral Formula for the Zero of a Continuous Function

Key Points

To provide an exact integral representation for the unique zero of a continuous function on a specific interval.
Demonstrated the continuity condition f(a) > 0 and f(b) < 0 indicating a zero in (a,b).
Derived a chain of equalities involving a double integral using the Fubini-Tonelli theorem.
Validated the findings through numerical methods.
Successfully established an integral representation of the zero within the defined conditions.
Validated the results with numerical examples confirming the theoretical approach.

Abstract

We establish an exact integral representation of the unique zero of a continuous function f on a,b satisfying f(a)>0, f(b)<0, and admitting exactly one zero in (a,b). The main result is the chain of equalities involving a double integral with two equivalent integration orders, related by the Fubini-Tonelli theorem. A numerical validation is provided.

An Integral Formula for the Zero of a Continuous Function

Key Points

Abstract

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