Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
Check continuity and one-sided slopes to distinguish a smooth point, corner, cusp, or vertical tangent.
College Board Topic 2.4 · CED p. 58 (PDF p. 63) · CED effective Fall 2020; project course outline 2026–27.
Differentiability at a point implies continuity there. To see the connection, write f(a+h)−f(a)=h·[f(a+h)−f(a)]/h for h≠0. If the quotient approaches a finite derivative, the product approaches 0, so the nearby values approach f(a). A discontinuity therefore rules out differentiability.
Continuity does not imply differentiability. For |x| at 0, left secant slopes are −1 and right secant slopes are 1, giving a corner. For x^(1/3) at 0, the difference quotient is h^(−2/3), which is unbounded positive: the graph has a vertical tangent, but no finite derivative. A cusp can have unbounded slopes of opposite signs.
To decide differentiability at a piecewise boundary, check continuity first, then compare the one-sided limits of the difference quotient based at the actual target value. Matching formulas for derivatives away from the boundary do not rescue a discontinuous target value. A vertical tangent is geometrically meaningful while its slope is not a finite real number.
Let's work through it
- For f(x)=|x| at 0, f is continuous because |x|→0=f(0). Reason: the continuity conditions hold.
- The left difference quotient |h|/h equals −1 and the right equals 1. Reason: the sign of h changes the quotient.
- The derivative does not exist. Reason: the two finite one-sided limiting slopes disagree.
Try it from memory
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Question 1
Can a function be differentiable at a point where it is discontinuous?
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Retrieval 1
Can a function be differentiable at a point where it is discontinuous?
Worked answer and reason
No.
Differentiability implies continuity.
Retrieval 2
Does a continuous graph with a corner have to have a derivative at the corner?
Worked answer and reason
No.
The two one-sided limiting slopes may disagree.
Catch the tempting mistake
- A graph without a hole must be differentiable.
- Corners, cusps and vertical tangents can occur in continuous graphs.
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Put it to work
- Unit 2 problem 11 · No calculator · analytic
- Unit 2 problem 12 · No calculator · graphical
- Unit 2 problem 13 · No calculator · analytic
For the teacher
- Compare a hole, jump, corner and vertical tangent using the finite derivative definition.
- Require a logical implication in the correct direction; use a counterexample for the converse.
Glossary and deeper reading · 6 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Differentiability
Differentiability means that a function has a finite derivative at the point under discussion. Differentiability at a point implies continuity there, but continuity alone does not imply differentiability. For example, |x| is continuous at 0 but its left and right tangent slopes differ, so it is not differentiable there. Corners, cusps, vertical tangents, and discontinuities can prevent a finite derivative.
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Continuity
A function is continuous at an input when its value exists, its finite limit exists, and those two values agree. At a domain endpoint, continuity uses the relevant one-sided approach. For example, x² is continuous at 3 because both its value and its nearby limit are 9. Continuity on an interval requires the appropriate condition at every input in that interval.
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Difference quotient
A difference quotient divides a change in a function’s output by the associated nonzero input change. The form [f(a+h)−f(a)]/h is an average rate over an interval of length h. For example, for f(x)=x² it simplifies to 2a+h when h≠0. Taking its limit as h approaches 0 defines the derivative at a if the finite limit exists.
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Vertical tangent
A vertical tangent is a tangent line parallel to the vertical axis, with no finite slope. A graph can have a vertical tangent even though its derivative does not exist as a finite real number at that input. For example, y=∛x has the vertical tangent x=0 at the origin as its nearby slopes tend to positive infinity. This case is separate from the finite-slope tangent approximation available at a differentiable point.
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One-sided limit
A one-sided limit uses inputs approaching a target from just one side. A left-hand limit uses smaller inputs, and a right-hand limit uses larger inputs. For example, a function that equals 0 left of 2 and 1 right of 2 has different one-sided limits at 2. The two-sided limit exists as a finite real number only when both one-sided limits exist and agree.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.