This has been confusing me for awhile. Let's say f(x) is undefined at x=2 and thus f(x) is discontinious at x=2. And it's derivative is f'(x)=2X. Is the derivative still defined at x=2 or not, because f(x) is undefined at x=2? If it is defined, wouldn't this mean f(x) is differentiable at x=2 and thus f(x) has to be continuous at x=2 because of the theorem: If f(x) is differentiable at a then f(x) must be continuous at a. One last question. Why is (x+1)(x-1)/(x-1) still undefined when x=1? Even though the x-1 cancel out. Thanks in advance.