Calculus BC · CED Unit 2: Differentiation · 14 min read · Updated 2026-05-11
Connecting differentiability and continuity — AP Calculus BC
AP Calculus BC · CED Unit 2: Differentiation · 14 min read
1. Theorem: Differentiability Implies Continuity★★☆☆☆⏱ 4 min
The core result of this topic is a formal logical implication: if a function $f$ is differentiable at $x=a$, then $f$ must be continuous at $x=a$. This can be proven directly using limit laws, and its contrapositive is one of the most useful tools for quickly testing differentiability on the AP exam.
Exam tip: Always check for discontinuity first when testing differentiability. If you find a discontinuity, you can stop immediately and write your conclusion, saving 1-2 minutes on exam day.
2. Continuity Does Not Imply Differentiability★★★☆☆⏱ 5 min
While differentiability guarantees continuity, the converse is not true: continuity is a necessary, but not sufficient, condition for differentiability. A function can be fully continuous at $x=a$ but still fail to be differentiable there. There are four common cases of continuous non-differentiable functions:
Oscillating tangents: the limit of the difference quotient does not exist
On the AP exam, this concept is most commonly tested for piecewise functions that are continuous at their junction point: you must check if the left-hand derivative equals the right-hand derivative to confirm differentiability.
Exam tip: Do not just compare derivatives of the pieces and skip the continuity check. If the function is discontinuous, equal derivatives on each side do not make it differentiable.
3. Identifying Non-Differentiable Points From Graphs★★☆☆☆⏱ 3 min
A common AP multiple-choice question gives you the graph of $f(x)$ and asks you to count how many non-differentiable points exist in a given interval. To solve these, check for any of the four types of non-differentiable points directly on the graph.
Exam tip: Don't forget to count vertical tangents as non-differentiable. Most students remember corners and discontinuities but miss vertical tangents on graph problems.
4. Finding Constants for Differentiable Piecewise Functions★★★★☆⏱ 6 min
A common free-response question asks you to find unknown constants in a piecewise function such that the function is both continuous and differentiable at the junction point. Always follow the order: find constants for continuity first, then solve for differentiability.
Common Pitfalls
Why: Students assume matching derivatives imply the function is connected, skipping the required continuity check that is a necessary condition for differentiability.
Why: Students confuse the direction of the implication, incorrectly inverting the 'differentiability implies continuity' rule.
Why: Students confuse the limit of the function with the function being continuous at the point.
Why: Students forget that $|x|$ is the classic example of a continuous non-differentiable function, and mix up the implication direction.
Why: Students rush and default to plugging in the most common junction point, leading to an incorrect comparison.