Which of the following best explains why the function is continuous at but not…
MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Which of the following best explains why the function f(x)=3x is continuous at x=0 but not differentiable there?
(a) Because f(x) changes sign at x=0, the slope from the left is negative while the slope from the right is positive, creating a sharp corner that prevents differentiability even though the function is continuous.
(b) Applying the power rule gives f′(0)=0, but nearby derivative values are not 0, so the derivative is discontinuous at 0 although the function itself is continuous.
(c) The limit of f(x) as x→0 equals f(0), but the difference quotient xf(x)−f(0) grows without bound as x→0, giving an infinite slope (vertical tangent) and hence no derivative at 0.},{
(d) Because the cube root of a negative number is not allowed in the derivative limit, the left–hand derivative does not exist while the right–hand derivative does, so the function stays continuous but is not differentiable at 0.