Consider the function
f(x)=(x−1)(x2−4x+3)(x2−1)(x−4).
The numerator and denominator are given in completely factored form.
a. Determine the complete set of all real numbers at which f is not continuous. Present your answer as a union of distinct points.
b. Classify each discontinuity found in part (a) as removable, jump, or infinite. Justify every classification by evaluating the necessary one-sided and/or two-sided limits.
c. Rewrite f as a new function h obtained by "removing" every removable discontinuity while leaving the domain elsewhere unchanged. Give an explicit formula for h defined on its domain.
d. Prove that h is continuous at every x in its domain. Your proof should make direct reference to the formal definition of continuity and to your work in parts (a)–(c).