So I've been studying Jay Cummings's Real Analysis book, and I've encountered this problem:Use the derivative to find all values of $ a $ such that the following holds: $$|x-a| = (x-2)^2 $$He doesn't explicitly mention that it should hold for all values of $x$. Although the solution is trivial: $$ a = x \pm (x-2)^2 $$ I really do not understand how come one uses the derivative for this.The reader is assumed to only know notions such as continuity and elementary set theory.
As an update regarding this problem, it seems that it is indeed an error. The reader who wants to check the errors in Jay Cummings's Real Analysis book should look here: https://longformmath.com/analysis-errata