2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

Differentiability implies continuity

If a function is differentiable at a point, it must be continuous there. However, the converse is false — a function can be continuous at a point but not differentiable there. This one-way relationship is a key logical fact tested often on the AP exam.

Ways a derivative can fail to exist

A derivative fails to exist at points with a corner (sharp turn, mismatched one-sided slopes), a cusp (slopes approach +∞ and −∞), a vertical tangent (slope is undefined/infinite), or a discontinuity (jump, hole, or asymptote). Recognizing these graph features helps identify non-differentiable points quickly.

Checking differentiability analytically

To confirm differentiability at a point, verify the function is continuous there AND that the left-hand and right-hand derivatives are equal. Use the limit definition: lim(h→0) [f(a+h)−f(a)]/h must exist and be the same from both sides.

Key terms

Differentiability → Continuity

If f is differentiable at a point, it must be continuous there (but not vice versa).

Corner

A point where left-hand and right-hand derivatives exist but are unequal; function is continuous but not differentiable.

Cusp

A point where the slope approaches +∞ from one side and −∞ from the other; derivative does not exist.

Vertical tangent

A point where the slope of the tangent line is undefined (infinite); function is not differentiable there.

Discontinuity and differentiability

Any type of discontinuity (jump, removable, infinite) automatically means the function is not differentiable at that point.

Try an AP-style question

Which describes f(x) = |x − 3| at x = 3?

1. Both continuous and differentiable
2. Differentiable but not continuous
3. Neither continuous nor differentiable
4. Continuous but not differentiable