Introduction to Differentiability and Continuity

Welcome! So far in Unit 2, you have learned that the derivative represents the "instantaneous rate of change" or the slope of a tangent line at a specific point. But here is the big question: Can you always find a derivative?

The answer is no. Just like some roads have potholes or dead ends, some functions have points where the derivative simply does not exist. In this chapter, we will explore the special relationship between continuity (the road is connected) and differentiability (the road is smooth) and identify the specific "trouble spots" where derivatives fail.

The Golden Rule: Differentiability Implies Continuity

There is a very important theorem you need to memorize for the AP Exam: If a function is differentiable at a point, it MUST be continuous at that point.

Think of it this way: to have a slope at a specific point, the graph has to actually be there and be connected. You cannot find the slope of a tangent line if there is a hole in the graph!

The "One-Way Street" Warning: While differentiability guarantees continuity, the reverse is not always true. Just because a function is continuous (you can draw it without lifting your pencil) does not mean it is differentiable. A function can be connected but still have a "sharp turn" that makes the derivative fail.

Key Takeaway:

If \(f(x)\) is differentiable at \(x = c\), then \(f(x)\) is continuous at \(x = c\).

If \(f(x)\) is not continuous at \(x = c\), then it cannot be differentiable at \(x = c\).

When Derivatives Fail to Exist

In the AP Calculus AB curriculum, there are four primary reasons why a derivative might fail to exist at a point \(x = c\). Let’s break them down:

1. Discontinuity

As we just mentioned, if a function has a hole, a jump, or a vertical asymptote at \(x = c\), the derivative \(f'(c)\) does not exist. Analogy: You can't measure the steepness of a bridge if the bridge is broken.

2. A Sharp Turn (Corner or Cusp)

A derivative fails when the slope from the left side of a point does not match the slope from the right side. Example: The absolute value function \(f(x) = |x|\) at \(x = 0\). To the left of zero, the slope is \(-1\). To the right of zero, the slope is \(1\). Because \(-1 \neq 1\), the derivative at the exact corner \(x = 0\) is undefined.

3. Vertical Tangent Lines

If a graph becomes so steep that the tangent line is perfectly vertical, the slope is "infinite." Since a derivative must be a real number, we say the derivative does not exist at that point. Example: \(f(x) = \sqrt[3]{x}\) has a vertical tangent at \(x = 0\).

4. Vertical Cusps

Similar to a sharp turn, a cusp is where the graph curves sharply and the slopes on both sides approach infinity (one positive and one negative). These look like "points" on a curve, similar to the top of a bird's wing.

Quick Review: The Four "Deal-Breakers"

  • Discontinuity: A break in the graph.
  • Corner: A sharp change in slope (like a "V").
  • Cusp: An extreme sharp point where slopes approach infinity.
  • Vertical Tangent: The graph goes straight up and down momentarily.

How to Check for Differentiability Formally

On the AP Exam, you might be asked to determine if a piecewise function is differentiable at the point where the two pieces meet (let’s call it \(x = c\)). To do this, you must pass two tests:

Step 1: Check for Continuity

Is the function connected at \(x = c\)? Confirm that \(\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)\). If the limits aren't equal, the function is not continuous and therefore not differentiable. Stop here!

Step 2: Check for "Smoothness" (Equal Derivatives)

If the function is continuous, now check the slopes. Does the derivative from the left match the derivative from the right? Check if \(\lim_{x \to c^-} f'(x) = \lim_{x \to c^+} f'(x)\). If these "one-sided derivatives" are equal, the function is differentiable at \(x = c\).

Common Mistakes to Avoid

Mistake 1: Forgetting to check continuity first. Students often jump straight to taking the derivative of both parts of a piecewise function. Even if the slopes match, if there is a "jump" in the graph, the derivative still doesn't exist!

Mistake 2: Assuming all "curvy" graphs are differentiable. Always look closely for vertical tangents. Functions like \(f(x) = x^{1/3}\) look smooth but have an undefined slope at the origin.

Did You Know?

The concept of differentiability is why GPS navigation works so well! The paths programmed into your GPS use "splines"—mathematical curves that are guaranteed to be differentiable. This ensures that the suggested path doesn't have "sharp turns" that would be impossible for a car to navigate at high speeds.

Practice Tips for the Exam

  • Multiple Choice: If you see a graph with a sharp point or a hole, immediately identify it as "not differentiable."
  • Free Response: If you are asked to "Justify" why a function is differentiable, you must explicitly show that the limit from the left equals the limit from the right for both the function \(f(x)\) (continuity) and the derivative \(f'(x)\) (smoothness).
  • Calculator: Remember that while your calculator can find numerical derivatives, it might sometimes give a false answer at a sharp corner. Always rely on your algebraic rules first!

Summary of Key Concepts

Differentiable means the graph is Continuous AND Smooth.
Not Differentiable means the graph has a Hole/Jump, a Sharp Corner, or a Vertical Tangent.