Can I define some functions as groups ecxept continuous function? Can somebody give me other examples of function that are group?
Thank you.
A set of functions, with a given operation, may be a group. For example, the set of all polynomials, of degree n or less, for any positive integer n, with addition as operation is a group. The "identity" is the constant polynomial, p(x)= 0 for all x. The "additive inverse" of the polynomial $a_nx^n+ a_{n-1}x^{n-1}+ \cdot\cdot\cdot+ a_1x+ a_0$ is $-a_nx^n- a_{n-1}x^{n-1}- \cdot\cdot\cdot- a_1x- a_0$.
The set of all invertible functions is a group with "composition" as operation.