Mathematics

Problem

Consider the function

p(x)=\frac{|x-1|-(x-1)}{x^{2}-1},\qquad x\in\mathbb R,\;x\ne \pm1.

(a) Determine the domain of

p

. (b) Simplify

p(x)

separately for the intervals

x>1

and

x<1

. Show all algebraic steps. (c) Identify every real number at which

p

is not continuous and classify each discontinuity as removable, jump, or infinite, providing limit evidence. (d) Decide whether redefining

p

at any point could make the function continuous on

(-\infty,\infty)

. If so, state the required redefinition and justify your answer.