Thread: Some true of false statements(Check my answers)

A.) If f(x) is locally linear at x = 5, then f(x) is definately differentiable at x = 5. False
B.) If g(x) has a point of discontinuity at x = 2, then f '(2) cannot possibly exist. True

I'm pretty sure I am right, but I'm never too sure. Could you please explain why the answers are what they are? Thanks.

Locally linear, intuitively, means that, near 5, the function looks like a line, and, in fact, the tangent line exists there....the only tricky part is when the tangent line is vertical. Then the function looks like a line locally, but the derivative does not exist (since a vertical tangent has an undefined slope). Thus, the answer to the first question is false.

In order for a function to be differentiable at a point, it must be continuous there. Thus, if a function is not continuous at a point, it's derivative there cannot exist, and the second question is true.