Analytic expression

(formula)

The totality of operations to be performed in a certain sequence on the value of an argument and on the constants in order to obtain the value of the function. Every function in one unknown with not more than a countable number of discontinuities has an analytic expression involving only three operations (addition, multiplication, passing to the limit by rational numbers), performed not more than a countable number of times, starting from an argument and from the constants, e.g.

If there is at least one analytic expression describing a given function, there are infinitely many such expressions. Thus, the function which is identically equal to zero is expressed by the series

and from any analytic expression it is always possible to obtain another one which is identically equal to the first:

where is an arbitrary analytic expression.

References

[1]	N.N. Luzin, "Theory of functions of a real variable" , Moscow (1948) (In Russian)