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Differentiability
& Continuity

Explore the deep connection between differentiability and continuity: when derivatives exist, when they fail, and why it matters.

AP Calculus BC

What Is Continuity?

A function \( f \) is continuous at \( x = a \) if three conditions hold simultaneously: the function is defined, the limit exists, and they match.

1. Defined
\( f(a) \) exists
2. Limit Exists
\( \displaystyle\lim_{x \to a} f(x) \) exists
3. They Match
\( \displaystyle\lim_{x \to a} f(x) = f(a) \)

What Is Differentiability?

A function \( f \) is differentiable at \( x = a \) if the derivative exists there: the limit of the difference quotient is a finite number.

Geometrically, differentiability means the graph has a unique, non-vertical tangent line at \( x = a \).

Smooth vs Non-Smooth Graphs

Differentiable Tangent exists Not Differentiable Corner: no unique tangent

The Core Theorem

Why? Differentiability requires the limit \(\displaystyle\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}\) to exist. This forces \(\displaystyle\lim_{h\to 0}[f(a+h)-f(a)] = 0\), which is exactly the continuity condition.

The Logical Relationship

Continuous Functions Differentiable Functions \(|x|\) corners \(x^{1/3}\) vertical tangent

Four Ways Differentiability Fails

1. Discontinuity
Jump, removable, or infinite: if \( f \) isn't continuous, it can't be differentiable.
2. Corner
Left-hand and right-hand derivatives exist but differ. Example: \( f(x)=|x| \) at \( x=0 \).
3. Cusp
One-sided derivatives go to \( +\infty \) and \( -\infty \). Example: \( f(x)=x^{2/3} \) at \( x=0 \).
4. Vertical Tangent
Both one-sided derivatives go to \( +\infty \) (or both \( -\infty \)). Example: \( f(x)=x^{1/3} \) at \( x=0 \).

Visualizing Non-Differentiable Points

Discontinuity Corner Cusp Vertical Tangent Not continuous → Not differentiable Continuous, sharp turn Left slope ≠ Right slope Continuous, pointed tip Slopes → ±∞ Continuous, vertical slope Slope → ∞ from both sides

How to Check Differentiability

1
Check continuity first. If \( f \) is not continuous at \( x = a \), stop: it is not differentiable. Verify \(\lim_{x\to a}f(x)=f(a)\).
2
Compute the left-hand derivative. Evaluate \(\displaystyle\lim_{h\to 0^-}\frac{f(a+h)-f(a)}{h}\).
3
Compute the right-hand derivative. Evaluate \(\displaystyle\lim_{h\to 0^+}\frac{f(a+h)-f(a)}{h}\).
4
Compare. If both one-sided limits are equal and finite, \( f \) is differentiable at \( a \). Otherwise, it is not.

Example 1: \( f(x) = |x| \) at \( x = 0 \)

\(\text{slope} = -1\) \(\text{slope} = +1\) \(x = 0\)

Example 2: Piecewise Function

1
Continuity: Left: \(\lim_{x\to 1^-}x^2 = 1\). Right: \(\lim_{x\to 1^+}(2x-1)=1\). And \(f(1)=1\). ✓ Continuous.
2
Left derivative: \(\displaystyle\lim_{h\to 0^-}\frac{(1+h)^2-1}{h} = \lim_{h\to 0^-}(2+h) = \) 2
3
Right derivative: \(\displaystyle\lim_{h\to 0^+}\frac{2(1+h)-1-1}{h} = \lim_{h\to 0^+}\frac{2h}{h} = \) 2
4
Both equal 2. ✓ Differentiable at \( x = 1 \) with \( f'(1) = 2 \).

Example 3: \( f(x) = x^{1/3} \) at \( x = 0 \)

vertical tangent \(x = 0\)

Common Mistakes to Avoid

✕ Converse Error
Assuming "continuous implies differentiable." Continuity is necessary but not sufficient. Always check one-sided derivatives.
✕ Skipping Continuity
Jumping straight to derivative computation without first verifying continuity. If \( f \) is discontinuous, stop immediately.
✕ Ignoring One-Sided Limits
Computing only \( f'(a) \) using a formula valid on one side. At boundary points, you must check both left and right derivatives separately.

Continuity vs Differentiability

Continuous at \(a\) Differentiable at \(a\)
Definition \(\lim_{x\to a}f(x) = f(a)\) \(\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}\) exists, finite
Geometric meaning No break in the graph Unique non-vertical tangent
Implies the other? Does NOT imply differentiability Implies continuity
\( f(x) = |x| \) at \(x=0\) ✓ Yes ✕ No (corner)
\( f(x) = x^{1/3} \) at \(x=0\) ✓ Yes ✕ No (vertical tangent)
\( f(x) = x^2 \) at \(x=0\) ✓ Yes ✓ Yes

Key Takeaways

1
Differentiable → Continuous is always true. Use the contrapositive: Not Continuous → Not Differentiable.
2
Continuous ↛ Differentiable. Corners, cusps, and vertical tangents are continuous but not differentiable.
3
Always check both sides. Compute left-hand and right-hand derivatives at piecewise boundaries and suspect points.
4
Finite and equal one-sided derivatives are required. If either is infinite or they differ, the derivative does not exist.

Question 1 of 5

If \( f \) is differentiable at \( x = 3 \), which of the following must be true?

Question 2 of 5

At which type of point is a function continuous but NOT differentiable?

Question 3 of 5

Let \( f(x) = \begin{cases} x^2\sin(1/x) & x \neq 0 \\ 0 & x = 0 \end{cases} \). Is \( f \) differentiable at \( x = 0 \)?

Question 4 of 5

Let \( g(x)=\begin{cases} 2x+1 & x < 2 \\ x^2-1 & x \geq 2 \end{cases} \). Is \( g \) differentiable at \( x = 2 \)?

Question 5 of 5

Which statement is the contrapositive of "If \( f \) is differentiable at \( a \), then \( f \) is continuous at \( a \)"?