AP® Calculus AB · Unit 2 · Topic 2.4
Connecting Differentiability and Continuity
Differentiability is the stronger condition: if a function has a derivative at a point, it must be continuous there. But continuity does not guarantee a derivative exists — corners, cusps, and vertical tangents break it.
A derivative at a point is a limit of slopes of secant lines as they close in on that point. If the function had a jump there, the secant slopes on either side would shoot off toward infinity or land on completely different values, and the limit would not exist. So for the derivative to exist at all, the function has to be unbroken there first — no jump, no hole, no vertical asymptote. That is why differentiable always forces continuous.
But continuity only guarantees the function has no break. It says nothing about the shape of the graph at that point. A graph can be perfectly connected and still have a sharp corner, a cusp, or a vertical tangent line, and at each of these the slope does not settle on one value. The limit definition falls apart not because the function jumps, but because the slope itself refuses to converge.
This is why Unit 1's three-part continuity test (the function value exists, the limit exists, and they match) is necessary but not sufficient for a derivative. Passing that test only clears the first hurdle. The absolute value function at x = 0 passes it easily: f(0) = 0, the limit from both sides is 0, they match. Differentiability demands a second, stricter test — do the one-sided slopes also match — and that is where corners fail.
Key terms
- Differentiable
- A function has a derivative at a point, meaning its difference quotient approaches one specific limiting slope.
- One-sided derivative
- The limit of the difference quotient taken from only the left or only the right of a point.
- Corner
- A point where the left-hand and right-hand slopes of a continuous graph disagree, blocking differentiability.
- Continuity
- A function has no jump, hole, or break at a point: the value, the limit, and their match all exist.
Watch it explained
- How it worksHow does the limit definition fall apart the moment the function jumps?
- Common mistakeWhy isn't every continuous function differentiable when the reverse always holds?
- Ties togetherHow does the three-part continuity test from Unit 1 explain why the absolute value function has no derivative at zero?
Check yourself
End of topic.