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.