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1. If there is only a single commodity type, then any weakly-monotonically increasing preference relation is convex. This is because, if , then every weighted average of y and ס is also . 2. Consider an economy with two commodity types, 1 and 2.
For example, if = is strictly increasing on the range [,], then it has an inverse = on the range [(), ()]. The term monotonic is sometimes used in place of strictly monotonic , so a source may state that all monotonic functions are invertible when they really mean that all strictly monotonic functions are invertible.
it is strictly increasing; it is s.t. () =. In fact, this is nothing but the definition of the norm except for the triangular inequality. Definition: a continuous function : [,) [,) is said to belong to class if:
It is often attached to a technical term to indicate that the exclusive meaning of the term is to be understood. The opposite is non-strict, which is often understood to be the case but can be put explicitly for clarity. In some contexts, the word "proper" can also be used as a mathematical synonym for "strict".
Explicitly, the map is called strictly convex if and only if for all real < < and all , such that : (+ ()) < + () A strictly convex function f {\displaystyle f} is a function that the straight line between any pair of points on the curve f {\displaystyle f} is above the curve f {\displaystyle f} except for the intersection points between the ...
The first-derivative test depends on the "increasing–decreasing test", which is itself ultimately a consequence of the mean value theorem. It is a direct consequence of the way the derivative is defined and its connection to decrease and increase of a function locally, combined with the previous section.
A real valued function () defined over an interval in the real line is called an absolutely monotonic function if it has derivatives () of all orders =,,, … and () for all in . [1]
Thus the probability density function of the normal distribution is a bell-curve, while the corresponding cumulative distribution function is a strictly increasing function that visually looks similar to a sigmoid function, which approaches 0 at −∞ and approaches 1 at +∞.