---
title: "AP Calculus 2.4: Differentiability and Continuity"
description: "Review AP Calculus Topic 2.4, including differentiability and continuity, when derivatives do not exist, corners, cusps, vertical tangents, discontinuities, one-sided derivatives, and piecewise functions."
canonical: "https://fiveable.me/ap-calc/unit-2/determining-when-derivatives-do-do-not-exist/study-guide/Lk6aXtIExtqciduDNGhk"
type: "study-guide"
subject: "AP Calculus AB/BC"
unit: "Unit 2 – Fundamentals of Differentiation"
lastUpdated: "2026-06-09"
---

# AP Calculus 2.4: Differentiability and Continuity

## Summary

Review AP Calculus Topic 2.4, including differentiability and continuity, when derivatives do not exist, corners, cusps, vertical tangents, discontinuities, one-sided derivatives, and piecewise functions.

## Guide

If a function is [differentiable](/ap-calc/key-terms/differentiable "fv-autolink") at a point, it must be [continuous](/ap-calc/key-terms/continuous "fv-autolink") there, but a continuous function is not always differentiable. Derivatives fail to exist at corners, cusps, vertical tangents, and any discontinuity. For AP Calculus, justify non-differentiability with a specific reason instead of only naming the point.

## Why This Matters for the AP Calculus Exam

This topic connects continuity from [Unit 1](/ap-calc/unit-1 "fv-autolink") to the idea of the derivative, and it shows up in both multiple-choice and free-response work. You will be asked to explain why a derivative does or does not exist, read graphs of functions to spot non-differentiable points, and test piecewise functions algebraically at a join. The skill of presenting a clear [difference quotient](/ap-calc/key-terms/difference-quotient "fv-autolink") or one-sided derivative structure is important for clear exam work, since a correct conclusion without supporting reasoning often will not support a stronger score. Getting comfortable with this now also sets up graph analysis in Unit 5, where you move between a function, its first derivative, and its second derivative.

## Key Takeaways

- Differentiable at a point means continuous at that point. The reverse is not guaranteed.
- If a point is not in the [domain](/ap-calc/key-terms/domain "fv-autolink") of $f$, it cannot be in the domain of $f'$.
- A derivative exists at a point only when the [one-sided limits](/ap-calc/key-terms/one-sided-limits "fv-autolink") of the difference quotient agree.
- Common ways differentiability fails: corners, cusps, vertical tangents, and discontinuities (jump, removable, or infinite).
- For piecewise functions, check continuity at the join first, then check that the left and right derivatives match.
- The [absolute value function](/ap-calc/key-terms/absolute-value-function "fv-autolink") $f(x)=|x|$ at $x=0$ and the cube root function $f(x)=\sqrt[3]{x}$ at $x=0$ are the classic examples of continuous but non-differentiable points.

## Continuity and Differentiability

Most curves you meet in AP Calculus are both continuous and differentiable, but some points break one or both conditions. If you need a refresher on continuity, check this guide: [Confirming Continuity over an Interval](/ap-calc/unit-1/confirming-continuity-over-interval/study-guide/HVxTuBB73RiPPODABBib).

For a function to be differentiable at a point, it must have a derivative there. Visually, if you zoom in close enough, the graph looks like a straight line. That line does not have to be horizontal.

Look at the graph of $\cos(x)$. As you zoom into the point $(0,1)$, the curve starts to look like a line.

###### Zooming into graph of $\cos(x)$

For a function to be differentiable at a point $x=a$, the [slope of the tangent line](/ap-calc/key-terms/slope-of-the-tangent-line "fv-autolink") must approach the same value from the left and the right:

$$\lim_{x\to a^{-}} f'(x) = \lim_{x\to a^{+}} f'(x) = f'(a)$$

Here is the key relationship: if a function is differentiable at a point, then it is continuous there. The reverse does not hold. A function can be continuous at a point and still fail to be differentiable. A function is not differentiable at a point if it is discontinuous, has a sharp change in [slope](/ap-calc/key-terms/slope "fv-autolink") (a corner or cusp), or has a vertical [tangent](/ap-calc/unit-2/derivatives-tangent-cotangent-secant-cosecant-functions/study-guide/wkBAhxiFZ1wV1M5mwLwt "fv-autolink").

A useful consequence: if a point is not in the domain of $f$, then it is not in the domain of $f'$. You cannot have a derivative where the function itself does not exist.

### Discontinuous Graphs

#### Unequal One-Sided Slopes

At most discontinuities, the slopes approaching from each side do not match.

For example, $g(x) = \frac{1}{x+2}$ has a domain of $(-\infty,-2) \cup (-2,\infty)$ because the denominator cannot be zero.

###### Graph of $g(x) = \frac{1}{x+2}$

Since this function is not continuous at $x=-2$ (it is not even defined there), it cannot be differentiable at $x=-2$.

#### Jump Discontinuity

The same reasoning applies to a [jump discontinuity](/ap-calc/key-terms/jump-discontinuity "fv-autolink"). The function value jumps from one level to another, so it is not continuous at that point and therefore not differentiable there.

###### Graph of a jump discontinuity

#### Removable Discontinuity

[Removable discontinuities](/ap-calc/key-terms/removable-discontinuities "fv-autolink") need a closer look. Consider $h(x) = \frac{x^2-4}{x-2}$. This simplifies to $x+2$ everywhere except $x=2$, where there is a hole.

###### Graph of a removable discontinuity

The derivative equals $1$ at every point except $x=2$. At $x=2$ the function is not defined, so the derivative does not exist there. Even though the slopes from both sides agree, the point is missing from the domain, so $h$ is not differentiable at $x=2$.

The takeaway: any kind of discontinuity at a point rules out differentiability at that point.

### Vertical Tangents

Look at the graph of $f(x)=2\left(x+2\right)^{\frac{1}{3}}$.

###### Graph with a vertical tangent

What is the derivative at $x=-2$? Think about the slope of the tangent line there. As you approach the point, the slope grows without bound toward $\infty$, because the change in $y$ stays nonzero while the change in $x$ shrinks toward $0$. A vertical line has no defined slope, so the derivative at $x=-2$ does not exist, and the curve is not differentiable there.

This matches the classic example $f(x)=\sqrt[3]{x}$ at $x=0$, where the [tangent line](/ap-calc/key-terms/tangent-line "fv-autolink") is vertical.

### Corners and Cusps

A function is not differentiable at a corner or a cusp because the slope does not approach the same value from both sides. Sometimes a cusp also involves a vertical tangent.

Look at the graph of $g(x) = |x|$. This function has a corner at $x=0$. Approaching from the left, the slope is $-1$; approaching from the right, the slope is $+1$. Since the one-sided slopes do not match, $g$ is not differentiable at $x=0$, even though it is continuous there.

###### Graph of $g(x)=|x|$ with a corner at the origin

This is one of the two classic examples worth memorizing: $|x|$ at $x=0$ (a corner) and $\sqrt[3]{x}$ at $x=0$ (a vertical tangent).

---

## How to Use This on the AP Calculus Exam

### MCQ

Many multiple-choice questions hand you a graph and ask where a function is not differentiable. Scan for:

- Breaks in the curve (jump, removable, or infinite discontinuities)
- Sharp corners where the slope flips suddenly
- Cusps where the curve comes to a point
- Vertical tangents where the slope blows up

Each of these spots fails differentiability, even if the function looks connected.

### Free Response

When a free-response part gives a [piecewise function](/ap-calc/key-terms/piecewise-function "fv-autolink") and asks whether it is differentiable at the join, use a clear two-step structure:

1. Check continuity at the join by matching the two pieces' values.
2. Check that the left-hand derivative and right-hand derivative are equal.

Showing this structure clearly is important for full credit. A correct yes or no with no supporting work usually will not earn the point.

### Problem Solving

Try this example. The function is

$$f(x) =  \begin{cases} \frac{8}{5}\sqrt{x+1}& x<3 \\ \frac{2}{5}x+2&  x\geq3\end{cases}$$

Given that $f(x)$ is continuous at $x=3$, is it differentiable there?

**Check the left-hand derivative.** Differentiate the piece for $x<3$:

$$f'(x) = \frac{d}{dx}\left(\frac{8}{5}\sqrt{x+1}\right) = \frac{4}{5\sqrt{x+1}}$$

So $\lim_{x\to 3^{-}} f'(x) = \frac{4}{5\sqrt{3+1}}= \frac{4}{5(2)} = \frac{2}{5}$.

**Check the right-hand derivative.** Differentiate the piece for $x\geq 3$:

$$f'(x) = \frac{d}{dx}\left(\frac{2}{5}x + 2\right) = \frac{2}{5}$$

So $\lim_{x\to 3^{+}} f'(x) = \frac{2}{5}$.

**Compare.** Both one-sided derivatives equal $\frac{2}{5}$, and the function is already continuous at $x=3$, so

$$\lim_{x\to 3^{-}} f'(x) = \lim_{x\to 3^{+}} f'(x) = \frac{2}{5}$$

The function is differentiable at $x=3$. The graph is smooth at that point, which confirms the result.

### Common Trap

Continuity does not prove differentiability. A piecewise function can match perfectly at the join (so it is continuous) but still have a corner there if the slopes do not match. Always check both conditions.

---

## Common Misconceptions

- "Continuous means differentiable." Not true. $|x|$ is continuous everywhere but not differentiable at $x=0$.
- "If the slopes match, the function must be differentiable." Only if the function is also continuous and the point is in its domain. A [removable discontinuity](/ap-calc/key-terms/removable-discontinuity "fv-autolink") can have matching one-sided slopes yet still fail because the point is missing.
- "A vertical tangent means the derivative is a huge number." A vertical tangent means the derivative does not exist, because a vertical line has no defined slope.
- "The derivative can exist wherever I want it to." If a point is not in the domain of $f$, it cannot be in the domain of $f'$.
- "Corners are differentiable as long as the graph is connected." A corner has different left and right slopes, so the derivative does not exist there even though the graph is continuous.
- "Removable discontinuities are fine for derivatives." The function is not defined at that hole, so it is not differentiable there.

## Related AP Calculus Guides

- [Unit 2 Overview: Differentiation](/ap-calc/unit-2/review/study-guide/HGq8OPntbFvqp1uRuEcj)
- [2.1 Defining Average and Instantaneous Rates of Change at a Point](/ap-calc/unit-2/defining-average-instantaneous-rates-change-at-point/study-guide/5ozgLc7Ucg4El3O0fV28)
- [2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions](/ap-calc/unit-2/derivatives-tangent-cotangent-secant-cosecant-functions/study-guide/wkBAhxiFZ1wV1M5mwLwt)
- [2.2 Defining the Derivative of a Function and Using Derivative Notation](/ap-calc/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD)
- [2.3 Estimating Derivatives of a Function at a Point](/ap-calc/unit-2/estimating-derivatives-function-at-point/study-guide/EzDxIj2eOHj3Ysb68Ew5)
- [2.7 Derivatives of cos x, sinx, e^x, and ln x](/ap-calc/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv)

## Vocabulary

- **continuity**: A property of a function at a point where the function is defined, the limit exists, and the limit equals the function value at that point.
- **derivative**: The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point.
- **difference quotient**: The expression [f(x+h) - f(x)]/h used to calculate the average rate of change and find the derivative as a limit.
- **differentiability**: A property of a function at a point where the derivative exists; a function is differentiable at a point if the limit of the difference quotient exists at that point.
- **domain**: The set of all input values (x-values) for which a function is defined.
- **left hand limit**: The value that a function approaches as the input approaches a point from values less than that point.
- **right hand limit**: The value that a function approaches as the input approaches a point from values greater than that point.
- **slope**: The steepness or rate of change of a line, calculated as the change in y-values divided by the change in x-values.
- **tangent line**: A line that touches a curve at a single point and has a slope equal to the derivative of the function at that point.

## FAQs

### What is the relationship between differentiability and continuity?

If a function is differentiable at a point, it must be continuous there. A continuous function, however, may still fail to be differentiable.

### When does a derivative not exist?

A derivative does not exist at discontinuities, corners, cusps, vertical tangents, or points where one-sided derivative limits do not agree.

### Can a function be continuous but not differentiable?

Yes. A function can be continuous but not differentiable at a corner, cusp, or vertical tangent, such as |x| at x = 0.

### How do you check differentiability of a piecewise function?

First check continuity at the join. Then compare the left-hand derivative and right-hand derivative. Both conditions must work.

### Why does a vertical tangent make the derivative fail?

A vertical tangent has no finite slope, so the derivative does not exist at that point.

### Does a removable discontinuity have a derivative?

No. If the function is not defined at the point, it is not differentiable there, even if nearby slopes appear to match.

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