AP Calculus AB and BC

Continuity vs Differentiability

Differentiability is the stronger condition: every differentiable function is continuous, but plenty of continuous functions are not differentiable. Continuity means no break in the graph; differentiability additionally means no corner, cusp, or vertical tangent.

Continuity

Use when: You need the graph to have no hole, jump, or asymptote at the point, which is the hypothesis for the IVT and the EVT.

Differentiability

Use when: You need a single well-defined tangent slope, which is the hypothesis for the MVT and for any derivative computation.

Side by side

	Continuity	Differentiability
Requires	$\lim_{x \to a} f(x) = f(a)$	The limit defining $f'(a)$ exists
Broken by	Holes, jumps, asymptotes	All of those, plus corners, cusps, vertical tangents
Implies the other	No	Yes, always
Counterexample	None needed	$f(x) = \lvert x \rvert$ at $x = 0$

The one-directional arrow is the whole comparison. If $f'(a)$ exists then $f$ must be continuous at $a$ , because a jump would make the difference quotient blow up. Nothing forces the reverse: a graph can arrive at a point from both sides with no break and still have two different slopes there.

The absolute value function is the permanent counterexample. At the origin it is continuous, since both one-sided limits and the function value all equal zero, but the slope is $-1$ on the left and $1$ on the right, so no single tangent slope exists.

For piecewise functions

Making a piecewise function differentiable at a seam takes two equations, not one. The pieces must meet, which is continuity, and their derivatives must match, which is differentiability. Solving only the first is the most common error.

Frequently asked questions

Can a function be differentiable but not continuous?

No. Differentiability at a point always implies continuity there, so a discontinuity rules out a derivative immediately.

Is a corner continuous?

Yes. A corner is continuous but not differentiable, because the one-sided derivatives are different finite numbers.

In the CED: Unit 1: Limits and Continuity, Unit 2: Defining the Derivative