Continuity and Differentiability

TL;DR

Continuity means a function's graph has no breaks or jumps, allowing you to draw it without lifting your pen. Differentiability means you can find a unique tangent line at every point, implying a smooth curve without sharp corners. These concepts are fundamental for understanding how functions behave and change.

1. The Mental Model

Imagine a function as a path on a graph. If you can walk along the path without ever needing to jump or step over a gap, it's continuous. If the path is also perfectly smooth, with no sharp turns or corners, it's differentiable.

2. The Core Material

What is Continuity?

A function $f(x)$ is continuous at a point $x=a$ if three conditions are met:
1. $f(a)$ exists: The function is defined at that point.
2. $\lim_{x \to a} f(x)$ exists: As you approach $a$ from both the left and right sides, the function values approach the same number. This is called the limit.
3. $\lim_{x \to a} f(x) = f(a)$: The limit you found is exactly equal to the function's value at that point.

Think of it like this: the point exists, the function agrees on where it's going (from both sides), and where it's going is exactly where it lands.

If any of these conditions fail, the function is discontinuous at $a$. Types of discontinuity include:
* Removable discontinuity (hole): The limit exists, but it doesn't equal $f(a)$ or $f(a)$ isn't defined.
* Jump discontinuity: The left and right-hand limits are different.
* Infinite discontinuity (vertical asymptote): The function goes off to infinity.

What is Differentiability?

A function $f(x)$ is differentiable at a point $x=a$ if its derivative $f'(a)$ exists at that point. The derivative at a point represents the instantaneous rate of change or the slope of the tangent line to the curve at that point.

For the derivative to exist at $x=a$, the following must be true:
1. The function must be continuous at $x=a$. This is a critical prerequisite. If it's not continuous, you can't draw a single, well-defined tangent.
2. The left-hand derivative must equal the right-hand derivative. This means the slope of the tangent line approaching from the left must be the same as the slope approaching from the right. If they're different, you'd have a sharp corner (like at the tip of a V-shape graph), and you can't draw a unique tangent.

Important Relationship:
* If a function is differentiable at a point, it must be continuous at that point.
* However, if a function is continuous at a point, it is not necessarily differentiable at that point. (e.g., $f(x) = |x|$ is continuous at $x=0$ but not differentiable there because of the sharp corner).

How to Check for Continuity and Differentiability

For Continuity:

Follow the three-step definition:
1. Calculate $f(a)$.
2. Calculate $\lim_{x \to a^-} f(x)$ (left-hand limit) and $\lim_{x \to a^+} f(x)$ (right-hand limit). If they're equal, the limit exists.
3. Compare $f(a)$ with the limit.

For Differentiability:

1. First, check for continuity at the point. If it's not continuous, it's not differentiable.
2. If it's continuous, then find the derivative $f'(x)$.
3. Calculate the left-hand derivative at $a$, $\lim_{h \to 0^-} \frac{f(a+h) - f(a)}{h}$.
4. Calculate the right-hand derivative at $a$, $\lim_{h \to 0^+} \frac{f(a+h) - f(a)}{h}$.
5. If these two derivatives are equal, the function is differentiable at $a$. Often, for piecewise functions, you can just differentiate each piece and then check if the left-sided derivative matches the right-sided derivative at the point where the definition changes.

3. Worked Example

Let's examine the function:
$f(x) = \begin{cases} 2x+3 & \text{if } x \le 2 \ 5x-3 & \text{if } x > 2 \end{cases}$

We'll check for continuity and differentiability at $x=2$.

1. Check for Continuity at $x=2$:

• Condition 1: $f(2)$ exists?
  Using the first piece of the function (since $x \le 2$), $f(2) = 2(2) + 3 = 4 + 3 = 7$. Yes, it exists.

• Condition 2: $\lim_{x \to 2} f(x)$ exists?

  • Left-hand limit: $\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (2x+3) = 2(2) + 3 = 7$.
  • Right-hand limit: $\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (5x-3) = 5(2) - 3 = 10 - 3 = 7$.
    Since the left-hand limit equals the right-hand limit, the limit exists and $\lim_{x \to 2} f(x) = 7$.

• Condition 3: $\lim_{x \to 2} f(x) = f(2)$?
  Yes, $7 = 7$.

Therefore, $f(x)$ is continuous at $x=2$.

2. Check for Differentiability at $x=2$:

Since it's continuous, we can proceed.

• Let's find the derivative for each piece:

  • For $x < 2$, $f'(x) = \frac{d}{dx}(2x+3) = 2$.
  • For $x > 2$, $f'(x) = \frac{d}{dx}(5x-3) = 5$.

• Left-hand derivative at $x=2$: This is the derivative of the first piece at $x=2$.
  $L f'(2) = 2$.

• Right-hand derivative at $x=2$: This is the derivative of the second piece at $x=2$.
  $R f'(2) = 5$.

Since $L f'(2)
e R f'(2)$ ($2
e 5$), the function is not differentiable at $x=2$.
This indicates a sharp corner at $x=2$.

4. Key Takeaways

• Continuity means a function's graph can be drawn without lifting your pen.
• A function is continuous at a point if its value exists, its limit exists, and they are equal.
• Differentiability implies continuity, but continuity does not imply differentiability.
• A function is differentiable at a point if it's continuous there and has a unique, well-defined tangent (no sharp corners or vertical tangents).
• To check for continuity, compare $f(a)$ with the left and right-hand limits at $a$.
• To check for differentiability, first ensure continuity, then compare the left and right-hand derivatives.

Common Mistakes to Avoid:
- Forgetting to check all three conditions for continuity.
- Assuming differentiability just because a function is continuous.
- Not checking both left-hand and right-hand derivatives (or limits for continuity) at the "joining" points of piecewise functions.
- Confusing the value of the derivative with the value of the function itself.

5. Now Try It

Determine if the function $g(x) = \begin{cases} x^2 & \text{if } x \le 1 \ x & \text{if } x > 1 \end{cases}$ is continuous and differentiable at $x=1$. Work through both tests systematically. Success looks like clearly stating "continuous and differentiable", "continuous but not differentiable", or "not continuous" with your reasoning for each.