5. TRUE/FALSE: If 𝑓(𝑥) is continuous at 𝑥 = 𝑎, then 𝑓′(𝑎) exists. Circle your response and give a brief explanation of your choice.

Answer 1

False.

Continuity does not imply differentiable.

For example consider f(x) = absolute value of x.

Then f(x) is continuous at x=0 but f'(0) doesn't exist because the left hand and right hand tangents at x=0 have fixed slopes -1. +1. For f'(0) to exist these slopes need to converge to the same value at x=0.