Connecting Differentiability and Continuity

Differentiability asks for more than continuity does. Continuity just means no breaks, holes, or jumps - you can trace the graph without lifting your pen. Differentiability asks that the graph also have a single, well-defined slope at that point. A graph can satisfy the first condition and fail the second if it has a sharp corner: the slope coming in from the left settles on one number, the slope coming in from the right settles on a different number, and since those two one-sided slopes never agree, the limit that defines the derivative does not exist. The pen never lifts, but it suddenly changes direction.

Take f(x) = |x - 2|. It is continuous everywhere, including at x = 2, since the left and right pieces meet at the same height, 0. But for x < 2, f(x) = -(x-2), with slope -1, and for x > 2, f(x) = x-2, with slope 1. As x approaches 2 from the left the difference quotient approaches -1; from the right it approaches 1. Two different one-sided limits means no derivative at x = 2, even though there's no break in the graph there. That corner is permanent - zooming in never smooths it out.

This is exactly why continuity is a necessary but not sufficient condition for differentiability: every differentiable function is automatically continuous, but the converse fails. Keep this in mind going forward, because these non-differentiable points do not disappear from the story. In Unit 5, when you hunt for critical points to find maxima and minima, you check every point where the derivative is zero or where it fails to exist at all - corners like this one are exactly the second kind, and ignoring them means missing real extrema.