Types of Discontinuities: Removable and Jump

TL;DR

Discontinuities are points where a function "breaks" or isn't connected. Removable discontinuities occur when there's a hole in the graph that could be "filled" by redefining a single point. Jump discontinuities happen when the function's value suddenly jumps from one point to another without connecting.

1. The Mental Model

Imagine you're drawing a continuous line without lifting your pen. A discontinuity is any point where you have to lift your pen. We're looking at two specific reasons why you might lift it.

2. The Core Material

When we talk about continuity, we mean a function whose graph you can draw without lifting your pen. When a function isn't continuous at a certain point, we call that a discontinuity. We'll focus on two main types: removable and jump.

Removable Discontinuity (Holes)

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A removable discontinuity is like a hole in your graph. The function is "almost" continuous at that point, but there's just a single point missing or misplaced. You could "fix" the discontinuity by simply defining or redefining the function's value at that single point.

Mathematically, a removable discontinuity at $x=c$ means:
1. The limit of the function as $x$ approaches $c$ exists. This means the function approaches the same value from both the left and the right.
2. However, either $f(c)$ is undefined, or $f(c)$ exists but doesn't equal the limit.

Think of it this way: both sides of the graph are heading towards the same y-value, but when they get there, either there's nothing, or there's a dot somewhere else.

A common way these appear is in rational functions where a factor in the numerator and denominator cancels out. This canceled factor indicates a "hole."

graph TD
    A["Limit exists at x=c?"] -->|Yes| B{"Is f(c) defined and equal to the limit?"}
    B -->|No| C["Removable Discontinuity (Hole)"]
    B -->|Yes| D["Function is continuous at x=c"]
    A -->|No| E["Other type of Discontinuity (e.g., Jump, Infinite)"]

Jump Discontinuity

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A jump discontinuity occurs when the function "jumps" from one y-value to another at a specific x-value. There's a clear break, and the function's value suddenly changes. You can't just fill in a single point to fix it; you'd need to fill in an entire vertical line segment.

Mathematically, a jump discontinuity at $x=c$ means:
1. The limit of the function as $x$ approaches $c$ from the left exists.
2. The limit of the function as $x$ approaches $c$ from the right exists.
3. However, these two one-sided limits are not equal.

This type of discontinuity often shows up in piecewise functions, where the definition of the function changes abruptly at a certain point.

3. Worked Example

Let's look at two functions:

Example 1: Removable Discontinuity

Consider the function $f(x) = \frac{x^2 - 4}{x - 2}$.

1. Domain: The denominator cannot be zero, so $x \neq 2$. This tells us there's a potential discontinuity at $x=2$.
2. Simplify: We can factor the numerator: $f(x) = \frac{(x-2)(x+2)}{x-2}$. For $x \neq 2$, we can cancel the $(x-2)$ terms, leaving $f(x) = x+2$.
3. Limit: As $x$ approaches 2, the limit of $x+2$ is $2+2=4$. So, $\lim_{x \to 2} f(x) = 4$.
4. Function Value: However, $f(2)$ is undefined because we can't plug $x=2$ into the original function.

Since the limit exists (it's 4) but $f(2)$ is undefined, there's a removable discontinuity at $x=2$. It's a hole at the point $(2, 4)$.

Example 2: Jump Discontinuity

Consider the piecewise function:
$g(x) = \begin{cases} x+1 & \text{if } x < 1 \\ x^2 & \text{if } x \ge 1 \end{cases}$

Let's check for discontinuity at $x=1$ where the definition changes.

1. Limit from the left: $\lim_{x \to 1^-} g(x) = \lim_{x \to 1^-} (x+1) = 1+1 = 2$.
2. Limit from the right: $\lim_{x \to 1^+} g(x) = \lim_{x \to 1^+} (x^2) = 1^2 = 1$.
3. Compare limits: Since the left-hand limit (2) is not equal to the right-hand limit (1), the overall limit as $x \to 1$ does not exist.

Because the one-sided limits exist but are different, there's a jump discontinuity at $x=1$. The function "jumps" from a y-value of 2 (approaching from the left) to a y-value of 1 (starting from the right).

4. Key Takeaways

• A removable discontinuity (a "hole") occurs when a limit exists but the function value at that point is either undefined or doesn't match the limit.
• You can often spot removable discontinuities in rational functions where a common factor cancels from the numerator and denominator.
• A jump discontinuity happens when the left-hand limit and the right-hand limit at a point both exist but are not equal.
• Jump discontinuities often appear in piecewise functions where the function's rule changes abruptly.
• For a removable discontinuity, you can imagine "plugging" the hole with a single point to make it continuous.
• For a jump discontinuity, there's a clear vertical gap between the function's values from the left and right.

Common Mistakes to Avoid:
- Don't assume that if a function is undefined at a point, it's automatically a jump discontinuity; it could be removable or even an infinite discontinuity.
- For piecewise functions, always check both the left and right limits at the switching point.
- For rational functions, canceling factors indicates a hole, not a vertical asymptote (which is an infinite discontinuity).
- Confusing a vertical asymptote with a jump discontinuity; vertical asymptotes involve limits going to infinity, while jump discontinuities have finite, but different, one-sided limits.

5. Now Try It

Consider the function $h(x) = \frac{x^2 - x - 2}{x - 2}$ and the piecewise function $k(x) = \begin{cases} 3x+1 & \text{if } x < 0 \\ 2x+1 & \text{if } x \ge 0 \end{cases}$. For each function, determine if it has a removable discontinuity or a jump discontinuity, and clearly explain your reasoning. What success looks like: identifying the type of discontinuity (if any) and showing the mathematical steps (like limits or factoring) that support your conclusion.