Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist - AP Calculus BC Study Guide

The Core Idea: Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

This topic establishes the fundamental relationship between two core concepts in calculus: continuity and differentiability. While a continuous function can be drawn without lifting your pencil, a differentiable function must also be "smooth" at every point. Differentiability is a more stringent condition than continuity. If a function is differentiable at a point, it is guaranteed to also be continuous at that point. However, the reverse is not true; a function can be continuous at a point but fail to be differentiable.

The central task is to identify precisely where and why a derivative might fail to exist. The derivative represents the instantaneous rate of change, or the slope of the tangent line. Therefore, a derivative fails to exist at any point where a unique, non-vertical tangent line cannot be drawn. This can happen for three specific reasons: the function has a break (discontinuity), the function has a sharp corner or cusp, or the function has a vertical tangent line. Understanding these conditions is crucial for analyzing the behavior of functions.

Key Definitions and Conditions

The differentiability of a function at a point is formally tied to the existence of a specific limit. A function $f$ is differentiable at $x=c$ if the following two-sided limit exists:

$$f^{\prime }(c)=\underset{h\to 0}{\lim }\frac{f(c+h)-f(c)}{h}$$

For this limit to exist, the limit from the left must equal the limit from the right. A derivative fails to exist when this condition is not met or when other foundational properties are violated. The reasons for a derivative failing to exist are:

1. Discontinuity: If a function $f$ is not continuous at a point $x=c$, then it cannot be differentiable at $x=c$. A break in the graph makes it impossible to define a single tangent line.

2. Unequal Left- and Right-Hand Derivatives: A function is not differentiable at a point if the left- and right-hand limits of the difference quotient are not equal. This means:

   $$\underset{h\to 0^{-}}{\lim }\frac{f(c+h)-f(c)}{h}\ne \underset{h\to 0^{+}}{\lim }\frac{f(c+h)-f(c)}{h}$$

   Graphically, this occurs at a sharp corner or a cusp. The slope approaching from the left is different from the slope approaching from the right.

3. Vertical Tangent: A function is not differentiable at a point where it has a vertical tangent line. At such a point, the slope is undefined. This corresponds to the case where the limit of the difference quotient approaches $+\infty$ or $-\infty$.

   $$\underset{h\to 0}{\lim }\frac{f(c+h)-f(c)}{h}=\infty \text{or}\underset{h\to 0}{\lim }\frac{f(c+h)-f(c)}{h}=-\infty$$

Understanding the Geometry of Non-Differentiability

The conditions for non-differentiability have clear graphical interpretations. Being able to connect the analytic rule to a visual feature is a critical skill.

• Discontinuities: If a graph has a hole, a jump, or a vertical asymptote at $x=c$, the function is not continuous at $c$. Since differentiability requires continuity, the function is automatically not differentiable at $c$. You cannot draw a tangent line at a point that doesn't exist or at a "jump."

• Corners and Cusps: These features are the graphical representation of unequal left- and right-hand derivatives.

  • A corner is a sharp point where the slopes on either side are finite but different. The classic example is the vertex of an absolute value function.

  • A cusp is a point where the slopes from the left and right both approach either $+\infty$ or $-\infty$, but from opposite directions. The graph forms an increasingly sharp point.

• Vertical Tangents: This occurs at a point where the graph becomes momentarily vertical. The tangent line at this point would be a vertical line, which has an undefined slope. Therefore, the derivative does not exist. Unlike a cusp, the slopes approaching from both the left and right go towards either $+\infty$ together or $-\infty$ together.

Core Concepts & Rules

• Differentiability Implies Continuity: If a function $f$ has a derivative at $x=c$, then $f$ must be continuous at $x=c$.

• Continuity Does Not Imply Differentiability: A function can be continuous at $x=c$ but fail to be differentiable at that point. The existence of a corner is a primary example.

• A derivative fails to exist at any point of discontinuity. This includes removable (hole), jump, and infinite (vertical asymptote) discontinuities.

• A derivative fails to exist where the graph has a sharp corner or cusp. This is because the slope from the left does not equal the slope from the right.

• A derivative fails to exist where the graph has a vertical tangent line. This is because the slope of a vertical line is undefined.

Step-by-Step Example 1: Analyzing a Piecewise Function

Consider the function $f(x)$ defined as:

$$f(x)=\{\begin{array}{ll}{x}^2+1& \text{if }x\le 2\\ 4x-3& \text{if }x>2\end{array}$$

Determine if $f(x)$ is differentiable at $x=2$.

Step 1: Check for continuity at $x=2$.

For $f$ to be differentiable at $x=2$, it must first be continuous at $x=2$. We must check if the function value equals the limit.

• Function Value:$f(2)=(2{)}^2+1=5$.

• Left-Hand Limit:${\lim }_{x\to 2^{-}}f(x)={\lim }_{x\to 2^{-}}(x^2+1)=(2{)}^2+1=5$.

• Right-Hand Limit:${\lim }_{x\to 2^{+}}f(x)={\lim }_{x\to 2^{+}}(4x-3)=4(2)-3=8-3=5$.

Since the left-hand limit, right-hand limit, and function value are all equal to 5, the function $f(x)$ is continuous at $x=2$. This means it might be differentiable.

Step 2: Check if the left- and right-hand derivatives are equal.

We need to find the derivative of each piece and evaluate the slope as $x$ approaches 2 from both sides.

• Derivative of the left piece (for $x<2$):$\frac{d}{dx}(x^2+1)=2x$.

• Derivative of the right piece (for $x>2$):$\frac{d}{dx}(4x-3)=4$.

Now, we evaluate the limit of the derivatives as $x$ approaches 2.

• Left-Hand Derivative: The slope approaching from the left is ${\lim }_{x\to 2^{-}}{f}^{\prime }(x)={\lim }_{x\to 2^{-}}(2x)=2(2)=4$.

• Right-Hand Derivative: The slope approaching from the right is ${\lim }_{x\to 2^{+}}{f}^{\prime }(x)={\lim }_{x\to 2^{+}}(4)=4$.

Step 3: Conclude based on the findings.

Because the left-hand derivative (4) is equal to the right-hand derivative (4), the derivative exists at $x=2$.

Conclusion: Yes, $f(x)$ is differentiable at $x=2$, and $f^{\prime }(2)=4$.

Step-by-Step Example 2: Analysis from a Graph

The graph of a function $g(x)$ is shown below. Identify all x-values on the interval $(a,k)$ where $g(x)$ is not differentiable. For each x-value, provide a justification based on the geometric properties of the graph.

(Imagine a graph with the following features: a jump discontinuity at x=c, a sharp corner at x=e, a vertical tangent at x=g, and a hole (removable discontinuity) at x=j.)

Solution:

The function $g(x)$ is not differentiable at $x=c,x=e,x=g,$ and $x=j$.

• At $x=c$: The derivative $g^{\prime }(c)$ does not exist because the function $g(x)$ is not continuous at $x=c$. The graph has a jump discontinuity, meaning ${\lim }_{x\to c^{-}}g(x)\ne {\lim }_{x\to c^{+}}g(x)$.

• At $x=e$: The derivative $g^{\prime }(e)$ does not exist because the graph has a sharp corner at $x=e$. This means the slope approaching from the left is not equal to the slope approaching from the right. Analytically, the left-hand and right-hand limits of the difference quotient are not equal.

• At $x=g$: The derivative $g^{\prime }(g)$ does not exist because the graph has a vertical tangent line at $x=g$. The slope of a vertical line is undefined. Analytically, the limit of the difference quotient would approach $\infty$ or $-\infty$.

• At $x=j$: The derivative $g^{\prime }(j)$ does not exist because the function $g(x)$ is not continuous at $x=j$. The graph has a removable discontinuity (a hole).

Using Your Calculator

The concepts of continuity and differentiability are primarily analytical and must be justified using definitions and theorems, not by calculator inspection alone. However, a graphing calculator is an excellent tool for visualizing potential issues and verifying your conclusions.

To investigate differentiability at a point $x=c$:

1. Graph the Function: Enter the function into Y1= and graph it. Use the ZOOM features to closely inspect the behavior of the graph around $x=c$. Look for visual cues of non-differentiability:

   • Is there a visible break or jump (discontinuity)?

   • Does the graph form a sharp corner?

   • Does the graph appear to become vertical?

2. Use the Numerical Derivative: The calculator can approximate the derivative at a point. To check for a corner, you can compare the numerical derivative just to the left and just to the right of the point of interest.

   • Go to MATH -> 8:nDeriv( or use the shortcut ALPHA -> F2 -> 3:d/dx(.

   • To check the left-hand derivative at $x=c$, evaluate the derivative at a very close point, such as $c-0.001$. The syntax would be: nDeriv(Y1, X, c-0.001).

   • To check the right-hand derivative, evaluate at c + 0.001. The syntax would be: nDeriv(Y1, X, c+0.001).

   • If the two resulting values are significantly different, it strongly suggests the existence of a corner, and thus the function is not differentiable at $x=c$.

Important: This numerical method is for exploration and verification only. It is not a valid mathematical justification on the AP Exam. An analytical argument based on the EKs is required.

AP Exam Quick Hit

Common Question Types

• Analyzing a Graph: Given the graph of $f(x)$, you will be asked to identify the x-values where $f^{\prime }(x)$ fails to exist and to justify your answer by naming the feature (e.g., "corner," "cusp," "discontinuity," "vertical tangent").

• Analyzing a Piecewise Function: You will be given a piecewise function and asked to determine if it is differentiable at the point where the definition changes. This requires a two-step check: first for continuity, then for the equality of the left- and right-hand derivatives.

• Conceptual True/False: A multiple-choice question might present several statements about the relationship between differentiability and continuity, asking you to identify the one that is true. For example: "If ${\lim }_{x\to a}f(x)$ exists, then $f$ is differentiable at $x=a$." (This is false).

Common Mistakes

• Assuming Continuity Implies Differentiability: This is the most frequent conceptual error. Students correctly establish that a function is continuous at a point and then incorrectly conclude that it must be differentiable. Always remember to perform the second step: comparing the left- and right-hand derivatives.

• Providing Vague or Incomplete Justifications: Simply stating "the derivative is undefined" is not sufficient. You must state why. The justification must be tied to one of the three conditions: the function is not continuous, the graph has a corner/cusp (unequal one-sided derivatives), or the graph has a vertical tangent.

• Confusing $f(c)$ and $f^{\prime }(c)$: When analyzing a piecewise function, students sometimes mistakenly set the two functions equal to each other ($f_1(c)=f_2(c)$) and think this checks for differentiability. This only checks for continuity. To check for differentiability, you must set the derivatives equal ($f_1^{\prime }(c)=f_2^{\prime }(c)$).

• Forgetting to Check for Continuity First: When analyzing a piecewise function, some students jump directly to taking the derivatives of the pieces and setting them equal. If the function is not continuous at that point to begin with, it cannot be differentiable, and the check on the derivatives is irrelevant.