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) $\displaystyle{X}$ , its codomain (the set of possible values) $Y$ , and a "rule" $\displaystyle{f}$ that assigns to every element $x\in X$ a unique element $y=f(x)\in Y$ .

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 $f(x)$ should be uniquely determined by $x$ . The following are examples of functions:

$f(x)=x ^ {2}+2x-2$

$f(x)=\sin{\log{x}}$

$f(x)=x^2$ for x>0, otherwise $f(x)= \sin{x}$

$f(x)=p(g(x))$

$g(x)=F'(x)$

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.

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Function