Question

if f is differentiable at a, must f be continuous at a? choose the correct answer below. a. no, if f is differentiable at a and does not have a vertical tangent at a, then f is continuous at a. b. no, if f is differentiable at a and positive at a, then f is continuous at a. c. no, if f is differentiable at a and does not have a corner at a, then f is continuous at a. d. yes, if f is differentiable at a, then f is continuous at a.

Explanation:

One of the fundamental results in calculus is that if a function is differentiable at a point, then it must be continuous at that point. Differentiability implies the existence of a well - defined tangent line at the point, which in turn requires the function to be unbroken (continuous) at that point. Options A, B, and C are incorrect because they wrongly suggest that there are additional conditions for a differentiable function to be continuous, when in fact differentiability alone guarantees continuity.

Answer:

D. Yes, if f is differentiable at a, then f is continuous at a.