Here's a figure that's not a function. Tangent lines at two points on the figure are drawn at a single value of x. If the derivative existed at this point which one would it be?

A figure could be constructed such that there are infinitely many different values for the derivative at a given point.

b) This is so fundamental I'm going to have to let you think about it yourself. How would you construct a tangent line to a point of discontinuity? If the limit of the function as x approaches some value doesn't exist can the limit of the difference quotient at that point exist?