Monotone and antitone functions
(not over ℝ just the domain you see =
These are examples of invertible functions.
Theorem on the inverse function of continuous strictly monotonic functions:
Suppose the function f:(a,b)→ℝ with -∞≤a<b≤+∞ is strictly increasing (resp., decreasing) and continuous. Let
lim f(x)=α≥-∞ and lim f(x)=β≤+∞ , if f is strictly increasing, resp.,
lim f(x) = β≤+∞ and limf(x ) = α≥-∞, if f is strictly decreasing,
Then f maps the interval (a,b) invertibly onto the interval (α,β). The inverse function f -1:(α,β)→(a,b) is also strictly increasing (resp., decreasing) and continuous, and we have:
lim(f -1)(y)=a and lim(f -1)(y)=b, if f is strictly increasing, resp.,
lim(f -1)(y)=b and lim(f -1 )(y)=a, if f is strictly decreasing.
Analogous statements hold for semi-open or closed intervals [a,b].[Source]
Also, If (m,n) ⊂ (a,b), the function f:(a,b)→ℝ, that is also true.
A history of 20th-century mathematics in ~500 words, with example pictures:
The inheritance of Cantor’s Set Theory allowed the 20th century to create the domain of “Functional Analysis”.
This comes about as an extension of the classical Differential and Integral Calculus in which one considers not merely a particular function (like the exponential function or a trigonometric function), but the operations and transformations which can be performed on all functions of a certain type.
The creation of a “new” theory of integration, by Émile Borel and above all Henri Lebesgue, at the beginning of the 20th century, followed by the invention of normed spaces by Maurice Fréchet, Norbert Wiener and especially Stefan Banach, yielded new tools for construction and proof in mathematics.
The theory is seductive by its generality, its simplicity and its harmony, and it is capable of resolving difficult problems with elegance. The price to pay is that it usually makes use of non-constructive methods (the Hahn-Banach theorem, Baire’s theorem and its consequences), which enable one to prove the existence of a mathematical object, but without giving an effective construction.
[I]n 1955 … [Nancy] … was the golden age of French mathematics, where, in the orbit of Bourbaki and impelled above all by Henri Cartan, Laurent Schwartz and Jean-Pierre Serre, mathematicians attacked the most difficult problems of geometry, group theory and topology.
New tools appeared: sheaf theory and homological algebra … which were admirable for their generality and flexibility.
The apples of the garden of the Hesperides were the famous conjectures stated by André Weil in 1954: these conjectures appeared as a combinatorial problem … of a discouraging generality….
Methods invented in topology to keep track of invariants under the continuous deformation of geometric objects, must be employed to enumerate a finite number of configurations.
Like Moses, André Weil caught sight of the Promised Land, but unlike Moses, he was unable to cross the Red Sea on dry land, nor did he have an adequate vessel. André Weil … was not … ignorant of these techniques …. But [he] was suspicious of “big machinery” ….
Homological algebra, conceived as a general tool reaching beyond all special cases, was invented by Cartan and Eilenberg (… in 1956). This book is a very precise exposition, but limited to the theory of modules over rings and the associated functors
“Tor”. It was already a vast synthesis of known methods and results, but sheaves do not enter into this picture. Sheaves … were created together with their homology, but the homology theory is constructed in an ad hoc manner….
In the autumn of 1950, Eilenberg …undertook with Cartan to [axiomatise] sheaf homology; yet the construction … preserves its initial ad hoc character.
When Serre introduced sheaves into algebraic geometry, in 1953, the seemingly pathological nature of the “Zariski topology” forced him into some very indirect constructions.
Holy crap! I’ve been so busy with school and work that I’ve been neglecting my blog. I don’t even remember the last time I reblogged something.