Function
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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. Lets call this function . A common notation to define is: . This tells us that is a function that squares it's argument (it's 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
Contents
[hide]Introductory Topics
Domain and Range
The domain of a function is the set of input values for the argument of a function. The range of a function is the set of output values for that function. For an example, consider the function . The domain of the function is the set where is a real number. The range is the set of all non-negative real numbers, because the square root can never return a negative value.
Real functions
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.
The graph of a function
Functions are often graphed To find out if a graph is a function, it must stand up to the vertical line test.
Intermediate Topics
Injections, surjections, bijections
- An injective function(or one-to-one) is a function which has distinct values for distinct arguments.
By definition, is injective if , or equivalently,
If and are finite sets injectivity implies .
Monotonic functions
A function is called monotonically increasing if holds whenever . If the inequality holds strictly then the function is called strictly increasing.
Similarlly, a function is called monotonically decreasing if holds whenever . If the inequality holds strictly then the function is called strictly decreasing.
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
Intuitively, a continuous function has the propriety that it's graph can be drawn without taking the pencil off the paper. But the reality about continuous function is more complex.
Epsilon-delta definition
A function is called continuous at if, for all , there exists such that and .
Heine definition
The previous definition of continuity at is equivalent with the following: for every sequence such that we have that .
It is easy to see that a function is continuous in isolated points, and is continuous in accumulation points iff the limit of the function in those point 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.
- ...
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
Integrability
Convexity
History of the concept
Without being used explicitly, the notion of function first appears at the ancient greeks and egyptians.
The rigurous 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.