Darboux's law of intermediate values

Target entity: Darboux's law of intermediate values
Context triple: [Johannes G. G. Darboux, notableWork, Darboux's law of intermediate values]

• A. Darboux theorem
  The Darboux theorem is a fundamental result in symplectic geometry stating that all symplectic manifolds are locally symplectomorphic to the standard symplectic space, implying that the symplectic form can always be put into a canonical local normal form.
• B. Denjoy–Young–Saks theorem
  The Denjoy–Young–Saks theorem is a result in real analysis that classifies the possible behaviors of the derivative of a real function at almost every point on the real line.
• C. Du Bois-Reymond theory of orders of infinity
  The Du Bois-Reymond theory of orders of infinity is a foundational framework in analysis that rigorously compares the growth rates of functions by classifying them into hierarchies of infinitesimal and infinite magnitudes.
• D. Stetigkeit und irrationale Zahlen
  "Stetigkeit und irrationale Zahlen" is Richard Dedekind’s seminal 1872 work in which he rigorously defines real numbers and continuity via Dedekind cuts, laying a foundation for modern analysis.
• E. Hilbert’s nineteenth problem
  Hilbert’s nineteenth problem is one of David Hilbert’s famous list of 23 problems, asking whether solutions to regular variational problems are always analytic.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Darboux's law of intermediate values
Target entity description: Darboux's law of intermediate values is a fundamental theorem in real analysis stating that the image of a continuous function on an interval contains every value between any two of its function values.

• A. Darboux theorem
  The Darboux theorem is a fundamental result in symplectic geometry stating that all symplectic manifolds are locally symplectomorphic to the standard symplectic space, implying that the symplectic form can always be put into a canonical local normal form.
• B. Denjoy–Young–Saks theorem
  The Denjoy–Young–Saks theorem is a result in real analysis that classifies the possible behaviors of the derivative of a real function at almost every point on the real line.
• C. Du Bois-Reymond theory of orders of infinity
  The Du Bois-Reymond theory of orders of infinity is a foundational framework in analysis that rigorously compares the growth rates of functions by classifying them into hierarchies of infinitesimal and infinite magnitudes.
• D. Stetigkeit und irrationale Zahlen
  "Stetigkeit und irrationale Zahlen" is Richard Dedekind’s seminal 1872 work in which he rigorously defines real numbers and continuity via Dedekind cuts, laying a foundation for modern analysis.
• E. Hilbert’s nineteenth problem
  Hilbert’s nineteenth problem is one of David Hilbert’s famous list of 23 problems, asking whether solutions to regular variational problems are always analytic.
• F. None of above. chosen