2.4 Differentiability & Continuity

One-way implication

If f is differentiable at a, then f is continuous at a. Differentiability is the stronger condition — a smooth curve with a well-defined tangent.

The converse fails

Continuous does NOT imply differentiable. A function can be continuous yet not differentiable at corners, cusps, vertical tangents, and any discontinuity (where it isn't even continuous).

Key terms

Differentiable ⇒ continuous

If f'(a) exists, f is continuous at a.

Converse is false

Continuous does not imply differentiable.

Corners & cusps

Continuous but not differentiable (no single tangent).

Vertical tangent

Derivative undefined (infinite slope).

Stronger condition

Differentiability is stronger than continuity.

Try an AP-style question

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

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