Mittevähenev

Mittevähenev is a mathematical term used to describe a relation or map that does not decrease as its input increases. In the standard setting, a function f defined on a totally ordered set D is mittevähenev if for all x and y in D, x ≤ y implies f(x) ≤ f(y). The same idea applies to sequences: a sequence (a_n) is mittevähenev if a_n ≤ a_{n+1} for every n. The term is commonly translated as nondecreasing or monotone increasing (not necessarily strictly increasing).

Examples and interpretation:

- The function f(x) = x on the real numbers is mittevähenev.

- The exponential function f(x) = e^x is mittevähenev on R.

- A constant function is mittevähenev.

- A function can be mittevähenev on a domain without being globally increasing, for instance f(x) = x^2 is mittevähenev on [0, ∞) but not on the entire R.

- A nondecreasing cumulative distribution function (CDF) in probability theory is a classic example of mittevähenev behavior.

Properties:

- The sum of two mittevähenevad functions is mittevähenev.

- If f is mittevähenev and g is nondecreasing, then the composition g∘f is mittevähenev on the appropriate domain.

- The property is preserved by pointwise maximum: max(f, g) is mittevähenev if f and g are.

Applications:

Mittevähenev functions appear in real analysis, probability (CDFs), economics (nondecreasing utilities), and order theory, where monotone structures are studied and compared.