A derivative is not always a slope. Consider thi example f: \R \rightarrow \{0,1\}:
f(x) = 0 if x is rational f(x) =1 if x is irrational
f is discontinuous at all rational x. It is continuous and differnetiable at all irrational x. It does not have a slope anywhere.
Is this correct? Source or proof? Ask Chris Connell.