Is the Commutator of x and f(x) Always Zero?

[prismaticcore]

Homework Statement

Is the commutator of x and any function of x zero?

Homework Equations

Taylor's theorem allows such a function to be expanded into polynomials, so that [f(x),x] may be expanded into terms of [x^n,x], which are all zero. Hence, f(x) and x commute.

The Attempt at a Solution

Is this a valid demonstration?

 

[kuruman]

You don't need a Taylor expansion. What do you think is x(f(x) - f(x)x?

 

[dextercioby]

This is true iff X and f(X) have the same domain, i.e. the domain of X is invariant.