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Let be a real-valued monotone function defined on an interval. Then the set of discontinuities of the first kind is at most countable.. One can prove [5] [3] that all points of discontinuity of a monotone real-valued function defined on an interval are jump discontinuities and hence, by our definition, of the first kind.
The term monotonic transformation (or monotone transformation) may also cause confusion because it refers to a transformation by a strictly increasing function. This is the case in economics with respect to the ordinal properties of a utility function being preserved across a monotonic transform (see also monotone preferences ). [ 5 ]
A function that is absolutely monotonic on [,) can be extended to a function that is not only analytic on the real line but is even the restriction of an entire function to the real line. The big Bernshtein theorem : A function f ( x ) {\displaystyle f(x)} that is absolutely monotonic on ( − ∞ , 0 ] {\displaystyle (-\infty ,0]} can be ...
The term removable discontinuity is sometimes broadened to include a removable singularity, in which the limits in both directions exist and are equal, while the function is undefined at the point . [a] This use is an abuse of terminology because continuity and discontinuity of a function are concepts defined only for points in the function's ...
Furthermore, the indirect utility function can be written as a linear function of wealth : (,,) = (,) which is a special case of the Gorman polar form. Hence, if all consumers have homothetic preferences (with the same coefficient on the wealth term), aggregate demand can be calculated by considering a single "representative consumer" who has ...
A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible. The set of discontinuities of any function must be an F σ set. If such a function existed, then the irrationals would be an F σ set.
Local nonsatiation (LNS [2]) is often applied in consumer theory, a branch of microeconomics, as an important property often assumed in theorems and propositions.Consumer theory is a study of how individuals make decisions and spend their money based on their preferences and budget.
In this way any monotone function can be written in a unique way as the sum of a continuous monotone function and a jump function. Since the formula for H ( x ) {\displaystyle H(x)} is a positive combination of characteristic functions, it is a uniformly convergent sum, so the analysis of Riesz & Sz.-Nagy (1990 , pp. 13–15) is particularly ...