Where Derivatives Fail to Exist

A derivative at a point is a limit: the slope of the secant line as the two points squeeze together, approached from the left and from the right. If those two one-sided slopes do not match, the limit does not exist, and neither does the derivative. Three shapes cause this mismatch in different ways. A corner has the slope jump suddenly between two finite values, like going from slope -1 to slope 1. A cusp has both one-sided slopes race off to infinity, but with opposite signs, so the graph snaps back on itself like a V with infinitely steep sides. A vertical tangent has both one-sided slopes go to infinity with the same sign, so the curve is smooth but the tangent line itself is vertical, and vertical lines have no defined slope.

This is why continuity is not enough to guarantee differentiability. Every one of these graphs can be perfectly continuous, no jumps, no holes, drawn in one stroke, and still have no derivative at that one point. A corner function like f(x) = |x| is continuous everywhere, but the direction the graph is heading changes instantly at x = 0. That instant change is exactly what the limit definition cannot tolerate.

This topic is the natural partner to the formal limit definition of the derivative introduced just before it: here you see concretely what it looks like when that limit fails. It also sets up everything after it, since every differentiation rule to come assumes the function is differentiable first — you need to be able to spot where that assumption breaks down.