A function has a removable discontinuity at but is otherwise continuous on
MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A function g has a removable discontinuity at x=1 but is otherwise continuous on (−2,5). After redefining g(1) to fill the hole, which statement best describes the differentiability of g at x=1?
(a) g′(1) can never exist since the function was originally undefined at x=1 before the redefinition.
(b) g′(1) exists only if the left-hand and right-hand limits of the difference quotient at x=1 are equal.
(c) g′(1) is guaranteed to exist because redefining g(1) makes g continuous at x=1.
(d) g′(1) does not exist because the difference quotient gives the indeterminate form 0/0 when x→1.