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

Introduction

students, this lesson explains one of the most important ideas in AP Calculus AB: how differentiability and continuity are connected, and how to tell when a derivative exists. A derivative measures how fast a function is changing at a point, like the speed of a car at an exact moment 🚗. But not every function has a derivative everywhere. In fact, before a function can be differentiable at a point, it must be continuous there.

Objectives

Explain what it means for a function to be differentiable and continuous.
Identify when a derivative exists and when it does not.
Use AP Calculus reasoning to analyze graphs and formulas.
Connect differentiability to the broader study of derivatives and differentiation.

As you study this topic, keep in mind that AP Calculus often asks you to justify whether $f'(a)$ exists using both algebra and graph behavior. The key is to understand not just how to compute derivatives, but also when derivative rules apply and when they do not.

Differentiability and Continuity: The Big Idea

A function $f$ is continuous at $x=a$ if three things are true: $f(a)$ is defined, $\lim_{x\to a} f(x)$ exists, and $\lim_{x\to a} f(x)=f(a)$. In everyday language, you can draw the graph at $a$ without lifting your pencil ✏️.

A function is differentiable at $x=a$ if its derivative $f'(a)$ exists. This means the function has a well-defined tangent slope at that point. If the graph has a sharp corner, a cusp, a vertical tangent, or a discontinuity, then the derivative does not exist there.

The most important relationship is this:

\text{If } f \text{ is differentiable at } a, \text{ then } f \text{ is continuous at } a.

However, the reverse is not always true. A function can be continuous and still not be differentiable. This is a major idea for AP Calculus AB and shows up often on graph analysis questions.

Why continuity is necessary

Differentiability requires a local slope. If the graph has a break, there is no single smooth direction to measure at that point. Since the function must be connected at the point to have a tangent line there, continuity is required.

For example, the absolute value function $f(x)=|x|$ is continuous at $x=0$, because the graph has no break. But it is not differentiable at $x=0$ because there is a sharp corner.

How to Tell When a Derivative Does Not Exist

There are several common situations where $f'(a)$ does not exist. Knowing these patterns helps you answer AP exam questions quickly and accurately.

1. Discontinuity

If $f$ is not continuous at $x=a$, then $f'(a)$ does not exist.

Example: let

$f(x)=\begin{cases}$

1, & x<2 \\

$3, & x\ge 2$

$\end{cases}$

At $x=2$, the graph has a jump. Since the function is discontinuous there, $f'(2)$ does not exist.

2. Corner or cusp

A corner happens when the left-hand slope and right-hand slope are different finite numbers. A cusp is similar, but the slopes may become infinitely steep in opposite directions.

Example: $f(x)=|x|$ has a corner at $x=0$.

To test differentiability, compare one-sided derivatives:

$\lim_{h\to 0^-} \frac{f(a+h)-f(a)}{h}$

\quad \text{and} \quad

$\lim_{h\to 0^+} \frac{f(a+h)-f(a)}{h}$

If these two limits are not equal, then $f'(a)$ does not exist.

For $f(x)=|x|$, the left-hand derivative at $0$ is $-1$ and the right-hand derivative at $0$ is $1$, so the derivative does not exist.

3. Vertical tangent

A vertical tangent means the slope becomes undefined because it grows without bound. If the graph becomes nearly vertical, the derivative may not exist as a finite number.

A classic example is $f(x)=\sqrt[3]{x}$ at $x=0$. The graph is continuous there, but the tangent line is vertical, so the derivative is undefined at that point.

4. Endpoint of a domain

At an endpoint, a two-sided derivative cannot be computed because the function is not defined on both sides.

For example, if $f(x)=\sqrt{x}$, then the domain starts at $x=0$. The derivative from the right may exist, but $f'(0)$ as a two-sided derivative does not exist because there is no left-hand side.

When a Derivative Does Exist

A derivative exists at $x=a$ when the graph is smooth enough near that point. Smooth means no breaks, corners, cusps, or vertical tangents, and the function behaves regularly on both sides.

A common AP idea is that many basic functions are differentiable wherever they are defined in their domains. For example:

Polynomials are differentiable for all real numbers.
Rational functions are differentiable wherever the denominator is not zero.
Exponential and logarithmic functions are differentiable on their domains.
Trigonometric functions are differentiable wherever they are defined.

For instance, if $f(x)=x^3-4x+1$, then $f'(x)=3x^2-4$ exists for every real number. There are no corners or breaks in the graph.

Using One-Sided Derivatives and Graphs

On the AP exam, you may be asked to determine differentiability from a graph or piecewise function. The strategy is to check continuity first, then compare one-sided slopes.

Suppose a function is defined by

$f(x)=\begin{cases}$

x+2, & x<1 \\

$3-x, & x\ge 1$

$\end{cases}$

To check differentiability at $x=1$:

Check continuity.

Left-hand value approaches $1+2=3$.
Right-hand value at $1$ is $3-1=2$.
Since the values do not match, the function is not continuous at $x=1$.

Since continuity fails, $f'(1)$ does not exist.

If the function were continuous, then you would compare the left and right derivatives. For a piecewise function, compute each side separately and see whether they match.

This is a powerful AP tool because many questions ask whether the derivative exists at a join point. The most common reason for failure is that the slopes do not match or the function is not continuous.

Differentiability as a Local Property

Differentiability is local, meaning it depends on what happens very close to the point $a$. A function might look smooth most of the time but fail at one specific location.

For example, the graph of $f(x)=\frac{|x|}{x}$ is not differentiable at $x=0$, but the issue is actually stronger: the function is not even defined at $x=0$. That means there is no derivative there.

Another important point is that differentiability is different from being "pretty" or "smooth-looking" on a graph. A graph can appear smooth at a small scale and still fail differentiability if the slope changes too abruptly. Always use the mathematical definition or test for one-sided behavior, not just appearance 👀.

AP Calculus Reasoning: How to Justify Your Answer

When AP Calculus asks whether $f'(a)$ exists, your reasoning should usually follow this order:

Check whether $f$ is continuous at $a$.
If not continuous, conclude $f'(a)$ does not exist.
If continuous, check for a corner, cusp, vertical tangent, or endpoint.
If the function is piecewise, compare one-sided derivatives.
If the graph is smooth and the slope is finite, then $f'(a)$ exists.

A good justification might sound like this: “The function is continuous at $x=2$, but the left-hand derivative and right-hand derivative are not equal, so $f'(2)$ does not exist.” That kind of explanation earns credit because it uses calculus language correctly.

Conclusion

students, the connection between differentiability and continuity is one of the core ideas in AP Calculus AB. If a function is differentiable at a point, then it must be continuous there. But continuity alone does not guarantee differentiability. To know whether $f'(a)$ exists, look for continuity first, then check for corners, cusps, vertical tangents, and endpoints. These ideas help you analyze graphs, piecewise functions, and real-world models with confidence.

Mastering this topic will make later differentiation rules easier to understand because you will know when a derivative formula is valid and when the derivative itself does not exist.

Study Notes

Differentiability at $x=a$ means $f'(a)$ exists.
If $f$ is differentiable at $a$, then $f$ is continuous at $a$.
Continuous does not always mean differentiable.
A derivative does not exist at a discontinuity.
A derivative does not exist at a corner, cusp, vertical tangent, or endpoint.
For piecewise functions, check continuity first, then compare one-sided derivatives.
Smooth graphs usually indicate differentiability, but always use mathematical evidence.
Polynomials are differentiable everywhere.
Rational functions are differentiable wherever the denominator is not zero.
On AP Calculus, clearly justify whether $f'(a)$ exists using continuity and slope behavior.