I've hit a snag in my studies, namely something my book labels "Corollary 10.1": Are there any other functions with the same derivative as [itex]x^2+2=2x[/itex]? You should quickly come up with several: [itex]x^2+3[/itex] and [itex x^2-4[/itex] for instance. In fact, [itex]d/dx[x^2+c]=2x[/itex] for any constant c. Are there any other functions, though, with the derivative 2x? Corollary 10.1 says that there are no such functions. Corollary 10.1: Suppose [itex]g'(x)=f'(x)[/itex] for all x in some open interval I, then for some constant c, [itex]g(x)=f(x) + c[/itex] for all x in the interval I. As I read it, the text completely contradicted itself. So where is my understanding broken? Also, I'm not sure why f'(x)=0 is undefined when x=|x|, since it's defined everywhere else. The absolute value of 0 is still 0, isn't it?