Google: Gemini 3 Flash Preview flagged on Does AI know AP Calculus AB?: "Flag — the core theorem and examples are correct, but the statement about endpoints is misleading. A function can be differentiable at an endpoint using a one-sided derivative, depending on the definition/context. Also, saying differentiability on the entire domain automatically rules out “vertical asymptotes” is imprecise because asymptotes may lie outside the domain. The central relationship is correct, but these inaccuracies make the response unsuitable as a fully reliable AP Calculus AB explanation."

In the context of AP Calculus AB, the relationship between differentiability and continuity is a fundamental concept that can be summarized by one key theorem and its important converse.

Here is the breakdown of how these two concepts interact:

### 1. The Fundamental Theorem: Differentiability Implies Continuity
If a function $f(x)$ is differentiable at a point $x = c$, then it **must** be continuous at $x = c$.

On the AP Exam, this is a powerful tool. If you are told a function is "differentiable on its entire domain," you can automatically assume there are no holes, jumps, or vertical asymptotes.

*   **Logical structure:** $D \implies C$ (If Differentiable, then Continuous).

### 2. The Converse is NOT Always True: Continuity does NOT imply Differentiability
Just because a function is continuous does not mean it is differentiable. A function can be a single, unbroken line but still fail to have a derivative at a specific point.

There are three common scenarios where a function is continuous but **not** differentiable:
*   **Sharp Turns (Corners/Cusps):** At a point like $x=0$ on $f(x) = |x|$, the graph is continuous, but the slope from the left ($m=-1$) does not match the slope from the right ($m=1$).
*   **Vertical Tangents:** If the graph becomes infinitely steep (like $f(x) = \sqrt[3]{x}$ at $x=0$), the derivative is undefined because the slope is infinite.
*   **Endpoints:** For a closed interval, a function is not differentiable at its endpoints because you cannot check the limit of the slope from both sides.

### 3. The Contrapositive: Non-Continuity Implies Non-Differentiability
This is the logical "flip side" of the first rule. If a function is **not continuous** at a point, it is **impossible** for it to be differentiable there.
*   If there is a hole, jump, or asymptote at $x = c$, $f'(c)$ does not exist.

### 4. How this appears on the AP Exam
You will likely encounter this relationship in three specific ways:

*   **Piecewise Functions:** You may be asked to find constants ($a$ and $b$) that make a function differentiable. To solve this, you must set up two equations:
    1.  **Continuity:** Set the two pieces equal to each other at the transition point.
    2.  **Differentiability:** Set the derivatives of the two pieces equal to each other at the transition point.
*   **Table Questions:** If a question states "The function $f$ is differentiable," the AP readers expect you to know this means $f$ is also continuous. You may need this to justify using the **Intermediate Value Theorem (IVT)** or the **Mean Value Theorem (MVT)**.
*   **Multiple Choice (The "True/False" trap):** A common distractor is an option that says "If $f$ is continuous, then $f'(c)$ exists." This is false (remember the absolute value graph).

### Summary Table
| If the function is... | Then it is... |
| :--- | :--- |
| **Differentiable** | Always Continuous |
| **Continuous** | Sometimes Differentiable (unless there's a sharp turn/vertical tangent) |
| **Not Continuous** | Never Differentiable |
| **Not Differentiable** | Could be Continuous (a corner) or Discontinuous (a hole) |