Can I take a derivative at x=0 for the function f(x) = |x| ?
Just because a function is continuous at a point, that doesn't mean it has a derivative at that point. If a function is differentiable on an interval, then it is continuous on that interval, but the reverse is not always true. If you draw a graph of the function f(x) = |x|, it's clear that we're going to have some problems at x=0. There's a sharp "kink" there, so how do we draw a tangent line? We could draw several, all of them grazing the point at some angle, but we don't know which one to pick. Can we just declare one of them the "correct" one, such as a horizontal line? You might think so, but it turns out that doesn't work either. We need to look at the difference quotient. The derivative of f(x) at a point a is defined by limh->0 of (f(a+h) - f(a))/(h). Substitute in f(x) = |x| and a=0, and we have limh->0 |h| / h. In order for this limit to exist, it has to approach the same value from both directions. If we approach from the negative direction, the numerator is always positive and the denominator is always negative, so the limit from the left is -1. If we approach from the positive direction, then both numerator and denominator are positive, so the limit from the right is +1. Since the left-limit and the right-limit aren't the same, the overall limit does not exist, so the derivative is undefined at x=0.
