Difference between revisions of "Function"
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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. | 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. | ||
| + | |||
| + | ==History of the concept== | ||
| + | |||
| + | ==Ways to define a function== | ||
| + | |||
| + | ==Injections, surjections, bijections== | ||
| + | |||
| + | ==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 <math> f:A\to B</math> is called [[monotonically increasing]] if <math>f(x_1)\geq f(x_2) </math> hols whenever <math>\displaystyle x_1>x_2 </math>. If the inequality holds strictly <math>(f(x_1)>f(x_2)) </math> | ||
| + | then the function is called [[strictly increasing]]. | ||
| + | |||
| + | Similarlly, a function <math> f:A\to B</math> is called [[monotonically decreasing]] if <math>f(x_1)\geq f(x_2) </math> hols whenever <math> x_1<x_2 </math>. If the inequality holds strictly <math>(f(x_1)>f(x_2)) </math> | ||
| + | then the function is called [[strictly decreasing]]. | ||
| + | |||
| + | ===The graph of a function=== | ||
| + | ===Continuity=== | ||
| + | ===Differentiability=== | ||
| + | ===Integrability=== | ||
| + | ===Convexity=== | ||
Revision as of 13:51, 19 June 2006
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 x>0, 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
History of the concept
Ways to define a function
Injections, surjections, bijections
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.
