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

Differentiability and Continuity at a Point

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

$$\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

That limit is $f'(a)$, the slope of the tangent line at $a$.

Think visually. If you zoom in very close to a differentiable point, the graph looks like a straight line. Not necessarily flat. Just smooth.

Here’s the relationship you must know cold:

• If a function is differentiable at $a$, then it is continuous at $a$.
• If a function is not continuous at $a$, then it is not differentiable at $a$.
• A function can be continuous but not differentiable at a point.

Also important:
If $a$ is not in the domain of $f$, then it cannot be in the domain of $f'$. No function value means no tangent line.

What Must Be True for a Derivative to Exist

To say $f$ is differentiable at $x = a$, all of this must happen:

1. $f(a)$ exists
2. $\lim_{x \to a} f(x)$ exists
3. The function is continuous at $a$
4. The left-hand and right-hand derivatives match

That last condition means:

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

If the one-sided derivatives are different, the derivative does not exist.

On FRQs, if they ask whether a function is differentiable at a point, you must justify both one-sided derivatives.

When a Function Is Not Differentiable

There are only a few ways this can fail. Learn to spot them instantly from a graph.

Discontinuities

Any discontinuity destroys differentiability.

That includes:

• Jump discontinuities
• Infinite discontinuities (vertical asymptotes)
• Removable discontinuities (holes)

Even if the slopes from both sides seem to approach the same number, a hole still means not continuous → not differentiable.

AP loves removable discontinuities where everything “almost” works. Don’t fall for it.

Corners

A corner happens when the slope from the left and right are finite but different. The graph of $f(x) = |x|$ is the classic example.

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

At $x = 0$:

• Left-hand slope < 0
• Right-hand slope > 0
• Not equal → derivative does not exist

Cusps

A cusp is sharper than a corner. The slopes blow up in opposite directions. A standard example is $f(x) = x^{2/3}$.

Cusp at $(0,0)$ for $f(x) = x^{2/3}$

• One side → $-\infty$
• Other side → $+\infty$

Derivative does not exist.

Vertical Tangents

The graph is continuous, but the tangent line is vertical.

For $f(x) = x^{1/3}$ at 0:

The slope approaches $\pm\infty$.
No finite slope → derivative does not exist.

This one shows up on multiple choice a lot.

Testing Differentiability for Piecewise Functions

This is extremely common on quizzes and FRQs.

Suppose

$$f(x) = \begin{cases} x^2 + 1 & x < 2 \\ 3x - 1 & x \ge 2 \end{cases}$$

To test differentiability at $x = 2$:

1. Check continuity

Left limit: $2^2 + 1 = 5$
Right value: $3(2) - 1 = 5$

Continuous.

2. Compare derivatives

Left derivative: $\frac{d}{dx}(x^2+1) = 2x \Rightarrow 2(2) = 4$
Right derivative: $\frac{d}{dx}(3x-1) = 3$

Since $4 \ne 3$, the function is not differentiable at 2.

Even though it’s continuous.

When you write this on an FRQ, you must explicitly evaluate both one-sided derivatives.

How This Gets Tested

From a graph, they’ll ask where $f'(x)$ does not exist. You scan for:

• Holes
• Jumps
• Asymptotes
• Corners
• Cusps
• Vertical tangents

From an equation, especially piecewise, they expect:

• Continuity check
• One-sided derivative comparison

Many students forget to check continuity first and lose easy points.