Calculus AB · Differentiation: Definition and Fundamental Properties · 14 min read · Updated 2026-05-10
Connecting differentiability and continuity — AP Calculus AB
AP Calculus AB · Differentiation: Definition and Fundamental Properties · 14 min read
1. The Core Theorem: Differentiability Implies Continuity★★☆☆☆⏱ 3 min
The first and most important result for this topic is the theorem that formalizes the relationship between the two properties: *If a function $f$ is differentiable at $x=a$, then $f$ must be continuous at $x=a$*. We can prove this directly from the definitions of both concepts.
Exam tip: If an AP FRQ asks for a constant to make a piecewise function differentiable, always solve for continuity first — AP graders award a point for this step, and it eliminates wrong answers in MCQ even if you don't finish the full problem.
2. The Converse Is Not True: Continuity Does Not Imply Differentiability★★★☆☆⏱ 4 min
The core theorem only guarantees that differentiability implies continuity; it does not work the other way around. A function can easily be continuous at a point but not differentiable at that point. This is the most commonly tested concept on the AP exam for this topic, as it probes whether students understand the difference between the two properties.
**Corner (or Kink)**: The function is continuous, but the one-sided derivatives from the left and right are different finite values. No single well-defined slope exists.
**Cusp**: The function is continuous, but the one-sided derivatives approach opposite infinities, so the slope is undefined.
**Vertical Tangent**: The function is continuous, but the tangent line is vertical, so the slope is infinite (undefined), hence the derivative does not exist.
Exam tip: Whenever you are asked to justify non-differentiability on an FRQ, you must explicitly compare one-sided derivatives or explain why the difference quotient limit is undefined — a conclusion without justification earns zero points.
3. Identifying Non-Differentiable Points From a Graph★★☆☆☆⏱ 2 min
One of the most common MCQ questions on this topic asks you to count or identify non-differentiable points from the graph of a function. You can do this without any algebra by looking for four key features, in order:
**Discontinuities**: Any jump, hole, infinite discontinuity, or mismatched point is automatically non-differentiable, by the contrapositive of the core theorem. Always count these first.
**Corners/Kinks**: A sharp turn in a continuous graph, where the left and right slopes are visibly different.
**Cusps**: A sharp point where the graph curves inward from both sides to meet at the point, with opposite infinite slopes.
**Vertical Tangents**: A point where the graph becomes vertical, so the slope is infinite (undefined) even though the function is continuous.
Exam tip: If the question asks for how many points, double-check that you didn't miss discontinuities — most students lose points here by only counting sharp turns and forgetting jumps/holes.
4. AP-Style Worked Practice Problems★★★★☆⏱ 5 min
Common Pitfalls
Why: Students remember the core theorem but forget it only works one direction, and the AP exam specifically tests this common misconception.
Why: If the function is discontinuous, the derivative cannot exist even if slopes match, leading to wrong values for the constants.
Why: Vertical tangents always occur at continuous points, so students confuse undefined slope with a break.
Why: Students assume that $\
Why: $\lim_{x \to a} f'(x) = f'(a)$, but this is only true if $f'$ is continuous at $a$, which is not guaranteed.
Why: Students confuse the limit of $f$ with the value of $f$, so they incorrectly assume continuity holds.