Function
The notion of a function is one of the basic notions of mathematics. To define a function, you need to know its domain (the set of admissible arguments) , its codomain (the set of possible values)
, and a "rule"
that assigns to every element
a unique element
.
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
When people talk of functions, most of the time they mean functions whose domain and codomain are sets of numbers (real or complex) or vectors. Since arbitrary functions can be arbitrarily bad and hard to handle, certain "good" classes of functions have been introduced like continuous functions, differentiable functions, measurable functions, etc. The functions in these classes possess many nice properties general functions don't have.
Contents
Ways to define a function
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
.
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.
Monotonic functions
A function is called monotonically increasing if
hols whenever
. If the inequality holds strictly
then the function is called strictly increasing.
Similarlly, a function is called monotonically decreasing if
hols whenever
. If the inequality holds strictly
then the function is called strictly decreasing.
The graph of a function
To find out if a graph is a function, it must stand up to the vertical line test.
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 XIXth 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 appeard, Mathematical Analysis, in which the notion of function has a central role.