|
|
| Line 16: |
Line 16: |
| | * [[Function/Introduction | Introduction to Functions]] | | * [[Function/Introduction | Introduction to Functions]] |
| | * [[Function/Intermediate | Intermediate Functions]] | | * [[Function/Intermediate | Intermediate Functions]] |
| − | * [[Function/Olympiad | Functions for Olympiad Students]] | + | * [[Function/Advanced | Functions for Olympiad and University Students]] |
| | | | |
| − |
| |
| | | | |
| − | ==Olympiad and University Level Topics==
| |
| | | | |
| − | ===Functions of Real Variables===
| |
| − | A real function is a function whose [[range]] is in the real numbers. Usually we speak about function whose domain is also a [[subset]] of the real numbers.
| |
| | | | |
| − | ====Continuity====
| + | == History of Functions == |
| − | Intuitively, a continuous function has the propriety that its graph can be drawn without taking the pencil off the paper. But the reality about continuous functions is more complex.
| |
| − | | |
| − | =====Epsilon-delta definition=====
| |
| − | A function <math>f:E\to\mathbb{R}</math> is called continuous at <math> x_{0} </math> if, for all <math> \varepsilon >0 </math>, there exists <math>\delta >0</math> such that <math> |x-x_0|<\delta </math> and <math>x\in E \Rightarrow |f(x)-f(x_0)|<\varepsilon </math>.
| |
| − | | |
| − | =====Heine definition=====
| |
| − | The previous definition of continuity at <math> x_{0} </math> is equivalent with the following: for every sequence <math> (x_n)_{n\geq 0} </math> such that <math> \displaystyle \lim_{n\to\infty}x_n=x_0 </math>, we have that <math> \lim_{n\to\infty}f(x_n)=f(x_0) </math>.
| |
| − | | |
| − | It is easy to see that a function is continuous in [[isolated point]]s, and is continuous in [[accumulation point]]s [[iff]] the limit of the function in those points equals the value of the function.
| |
| − | | |
| − | A function is continuous on a set if it is continuous in every point of the set.
| |
| − | | |
| − | =====Properties of continuous functions=====
| |
| − | *The sum and product of two continuous functions is a continuous function.
| |
| − | *The composition of two continuous functions is a continuous function.
| |
| − | * In a closed interval [a, b], there exist real numbers c and d such that f has a maximum value at c and f has a minimum value at d.
| |
| − | * Intermediate Value Property (see below)
| |
| − | | |
| − | =====Intermediate value property=====
| |
| − | If a function is continuous, then it has the [[Intermediate value property]]. The converse is not always true.
| |
| − | ''Proof'':...
| |
| − | | |
| − | =====Continuity on compact intervals=====
| |
| − | | |
| − | ====Differentiability====
| |
| − | | |
| − | For functions of one variable, differentiablility is simply the question of whether or not a derivative exists. For functions of more than one variable, it's significantly more complicated. In the case of both one and multivariable functions, differentiability implies continuity.
| |
| − | | |
| − | ====Integrability====
| |
| − | ====Convexity====
| |
| − | | |
| − | ==History of the concept== | |
| | Without being used explicitly, the notion of function first appears with the ancient Greeks and Egyptians. | | Without being used explicitly, the notion of function first appears with the ancient Greeks and Egyptians. |
| | | | |
This article is a stub. Help us out by expanding it.
A function is a rule that maps one set of values to another set of values. For instance, one function may map 1 to 1, 2 to 4, 3 to 9, 4 to 16, and so on. This function has the rule that it takes its input value, and squares it to get an output value. Let's call this function 
. A common notation to define is: 
. This tells us that is a function that squares its argument (its input value). Note that this "rule" can be arbitrarily complicated and doesn't need to be given by a simple formula or description. The only requirement is that 
should be uniquely determined by 
. The following are examples of functions:
for 
, otherwise 
Since functions cover such an enormous part of mathematics, we divide this topic into several articles:
History of Functions
Without being used explicitly, the notion of function first appears with the ancient Greeks and Egyptians.
The rigorous definition was stated in the 19th century and is the result of the works of some famous mathematicians: A.L. Cauchy, L. Euler, B. Riemann. With the development of set theory, a new branch of mathematics appeared, mathematical analysis, in which the notion of function has a central role.
See Also
. A common notation to define 
. This tells us that 
should be uniquely determined by 
. The following are examples of functions:
for 
, otherwise 
