Jensen functional equation

Target entity: Jensen functional equation
Context triple: [Cauchy functional equation, specialCaseOf, Jensen functional equation]

• A. Cauchy functional equation
  The Cauchy functional equation is a fundamental equation in functional analysis and real analysis, typically of the form f(x + y) = f(x) + f(y), whose solutions characterize additive functions and illustrate the contrast between regular (e.g., continuous) and highly pathological behaviors.
• B. Jensen inequality
  Jensen's inequality is a fundamental result in convex analysis and probability theory that relates the value of a convex (or concave) function of an expectation to the expectation of the function, providing bounds that underlie many other inequalities and convergence results.
• C. 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.
• D. Ulam stability
  Ulam stability is a concept in the theory of functional equations that studies when approximate solutions imply the existence of exact solutions nearby, forming the basis of what is now called Hyers–Ulam stability.
• E. Du Bois-Reymond function
  The Du Bois-Reymond function is a classic example of a continuous but nowhere differentiable function, illustrating pathological behavior in real analysis.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Jensen functional equation
Target entity description: The Jensen functional equation is a generalization in functional analysis that characterizes Jensen-convex functions and underlies Jensen’s inequality in convexity theory.

• A. Cauchy functional equation
  The Cauchy functional equation is a fundamental equation in functional analysis and real analysis, typically of the form f(x + y) = f(x) + f(y), whose solutions characterize additive functions and illustrate the contrast between regular (e.g., continuous) and highly pathological behaviors.
• B. Jensen inequality
  Jensen's inequality is a fundamental result in convex analysis and probability theory that relates the value of a convex (or concave) function of an expectation to the expectation of the function, providing bounds that underlie many other inequalities and convergence results.
• C. 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.
• D. Ulam stability
  Ulam stability is a concept in the theory of functional equations that studies when approximate solutions imply the existence of exact solutions nearby, forming the basis of what is now called Hyers–Ulam stability.
• E. Du Bois-Reymond function
  The Du Bois-Reymond function is a classic example of a continuous but nowhere differentiable function, illustrating pathological behavior in real analysis.
• F. None of above. chosen