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

For a function to be differentiable at a certain point, it must be continuous at that point. This is just a given for derivatives of functions (i.e. derivatives only exists at x values where the function is continuous) Therefore, if f(x) is continuous at x = a, then the derivative f '(x) must also exist at x = a by definition.