Topic 2.4 Notes – Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

What Differentiability Means at a Point

A function is differentiable at $x=a$ if this limit exists and is finite:

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

That limit is the slope of the tangent line at $x=a$.

Geometrically, if you zoom in close enough to the graph near that point, it looks like a straight, non-vertical line.

Another way to think about it:

• The left-hand derivative and right-hand derivative must both exist
• They must be equal
• The slope must be a real number (not ±∞)

The Big Relationship

Here’s the rule you must know cold:

• If $f$ is differentiable at $a$, then $f$ is continuous at $a$.
• If $f$ is not continuous at $a$, it cannot be differentiable there.
• A function can be continuous and still not differentiable.

So:

$$\text{Differentiable} \Rightarrow \text{Continuous}$$ $$\text{Continuous} \not\Rightarrow \text{Differentiable}$$

Also important: if $a$ is not in the domain of $f$, then $a$ is not in the domain of $f'$.

When a Function Is Continuous but Not Differentiable

There are two main AP-level reasons this happens.

Corners and Cusps

At a corner, the graph changes direction sharply. The slopes from the left and right don’t match.

Classic example:
$$f(x) = |x|$$

Corner at $x=0$ for $f(x)=|x|$

Focus on the point at $x=0$. The graph is continuous there, but the left and right slopes are different.

At $x=0$:

• Left-hand slope = −1
• Right-hand slope = +1
• Since they are not equal, $f'(0)$ does not exist.

On tests, this often appears in piecewise functions. You’ll:

1. Take derivative of each piece.
2. Plug the point into the left derivative.
3. Plug into the right derivative.
4. Compare.

If they don’t match, you’ve found a corner.

A cusp is similar, except slopes may go to ±∞ with opposite signs. Same conclusion: derivative does not exist.

Vertical Tangents

Sometimes the graph is smooth but becomes infinitely steep.

Example:
$$f(x) = x^{1/3}$$

Vertical tangent at $x=0$ for $f(x)=x^{1/3}$

Near $x=0$, the curve is smooth, but the slope grows without bound and the tangent line becomes vertical.

The key idea:
A derivative must be a finite number. If the slope approaches ±∞, the derivative does not exist.

Students sometimes say “the derivative is infinity.” On the AP exam, that’s incorrect. The derivative does not exist.

Discontinuity Automatically Kills Differentiability

If a function is not continuous at $x=a$, it is not differentiable there.

Types you should instantly recognize from a graph:

• Jump discontinuity
• Removable discontinuity (a hole)
• Infinite discontinuity (vertical asymptote)

Why this works: the derivative definition depends on a limit. If the function itself breaks, the limit in the derivative cannot work.

Even if slopes from both sides look equal near a hole, if the function value at that point is missing or incorrect, the derivative does not exist there.

Piecewise Functions at a Boundary Point

This is extremely common on quizzes and FRQs.

To check differentiability at $x=a$:

1) Check continuity

• Compute left-hand limit.
• Compute right-hand limit.
• Make sure they equal $f(a)$.

If it’s not continuous, stop. Not differentiable.

2) Compare one-sided derivatives

• Differentiate each formula.
• Evaluate left derivative at $a$.
• Evaluate right derivative at $a$.
• If equal → differentiable.
• If not → corner → not differentiable.

On some FRQs, they’ll tell you the function is continuous. That’s your signal to skip straight to comparing derivatives.

Reading a Graph Quickly

If they ask, “At how many points does $f'$ fail to exist?” scan for:

• Breaks in the graph
• Sharp turns
• Vertical tangents

Do not count:

• Smooth local maxima or minima
• Horizontal tangents

A horizontal tangent just means slope = 0. That derivative exists.

When you look at a smooth curve with no breaks, no sharp points, and no infinite steepness, the derivative exists there.