Introduction to Continuous Functions

TL;DR

Continuous functions are like drawing without lifting your pencil; there are no breaks, jumps, or holes in their graphs. Mathematically, a function is continuous at a point if its value at that point matches the limit as you approach it. Understanding continuity is key for calculus, as many important theorems only apply to continuous functions.

1. The Mental Model

Imagine you're drawing a function's graph. If you can draw the entire thing without ever lifting your pencil off the paper, that function is continuous. If you have to lift your pencil—say, to skip a gap or jump to a new level—then it's not continuous.

2. The Core Material

A function $f(x)$ is continuous at a specific point $x = c$ if three conditions are met:

1. $f(c)$ is defined: The function must actually have a value at that point. You can't have a hole there.
2. $\lim_{x \to c} f(x)$ exists: As you get closer and closer to $c$ from both the left and the right, the function's value must approach a single, specific number. This means no jumps or breaks.
3. $\lim_{x \to c} f(x) = f(c)$: The value the function approaches (the limit) must be exactly the same as the function's actual value at that point. This links the limit and the function value, preventing "holes" where the limit exists but the point itself is elsewhere.

If a function is continuous at every point in its domain, we just say it's a continuous function. Most of the functions you've worked with so far (polynomials, sines, cosines, exponentials) are continuous everywhere.

Types of Discontinuities

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When a function isn't continuous, we call it discontinuous, and the points where it breaks are called discontinuities. There are a few main types:

• Removable Discontinuity (Hole): This happens when the limit exists at a point, but either $f(c)$ isn't defined or $f(c)$ is defined somewhere else. Imagine a single point missing from your drawn line, or a single dot floating above or below the line. You could "remove" it by just defining or redefining $f(c)$ to match the limit.

  • Example: $f(x) = \frac{x^2 - 1}{x - 1}$. For $x=1$, you'd divide by zero. But you can simplify it to $f(x) = x+1$ for $x \neq 1$. There's a hole at $x=1$.

• Jump Discontinuity: This occurs when the limit from the left side of $c$ is different from the limit from the right side of $c$. The graph literally "jumps" from one value to another.

  • Example: Piecewise functions like $f(x) = \begin{cases} x+1 & \text{if } x < 0 \\ x-1 & \text{if } x \ge 0 \end{cases}$. At $x=0$, the function jumps.

• Infinite Discontinuity (Vertical Asymptote): This happens when the function's value shoots off to positive or negative infinity as $x$ approaches $c$.

  • Example: $f(x) = \frac{1}{x}$. At $x=0$, the function approaches infinity from one side and negative infinity from the other.

Here's how these types relate:

graph TD
    A["Function is Discontinuous at x=c"] --> B{"Does limit as x->c exist?"}
    B -- Yes --> C["Removable Discontinuity (Hole)"]
    B -- No --> D{"Are left-hand and right-hand limits finite but different?"}
    D -- Yes --> E["Jump Discontinuity"]
    D -- No --> F{"Does at least one limit go to +/- infinity?"}
    F -- Yes --> G["Infinite Discontinuity (Vertical Asymptote)"]
    F -- No --> H["Other (e.g., oscillating discontinuity)"]

3. Worked Example

Let's look at the function $f(x) = \begin{cases} x^2 & \text{if } x < 2 \\ 5 & \text{if } x = 2 \\ x+2 & \text{if } x > 2 \end{cases}$. We want to determine if $f(x)$ is continuous at $x=2$.

We need to check the three conditions:

1. Is $f(2)$ defined?
   From the definition, when $x=2$, $f(x)=5$. So, $f(2)=5$. Yes, it's defined.

2. Does $\lim_{x \to 2} f(x)$ exist?
   We need to check the left-hand and right-hand limits:

   • Left-hand limit: As $x$ approaches 2 from values less than 2 ($x<2$), we use the $x^2$ part of the function:
     $\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} x^2 = (2)^2 = 4$.
   • Right-hand limit: As $x$ approaches 2 from values greater than 2 ($x>2$), we use the $x+2$ part of the function:
     $\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (x+2) = (2)+2 = 4$.
     Since the left-hand limit ($4$) equals the right-hand limit ($4$), the overall limit exists and $\lim_{x \to 2} f(x) = 4$.

3. Does $\lim_{x \to 2} f(x) = f(2)$?
   We found $\lim_{x \to 2} f(x) = 4$ and $f(2) = 5$.
   Since $4 \neq 5$, the third condition is not met.

Therefore, the function $f(x)$ is not continuous at $x=2$. It has a removable discontinuity because the limit exists, but the function's value at that point is different.

4. Key Takeaways

• A function is continuous if you can draw its graph without lifting your pencil.
• Mathematically, continuity at a point means $f(c)$ is defined, $\lim_{x \to c} f(x)$ exists, and these two values are equal.
• Most basic functions like polynomials, trig functions, and exponentials are continuous over their natural domains.
• Discontinuities can be holes (removable), jumps (different one-sided limits), or vertical asymptotes (infinite limits).
• Continuity is a fundamental concept for understanding derivatives and integrals in calculus.

Common mistakes to avoid:
- Forgetting to check all three conditions for continuity at a point.
- Mixing up a limit that doesn't exist (like a jump) with a limit that exists but doesn't match the function value (a hole).
- Assuming a function is continuous everywhere just because its components are; always check where the function's definition changes.
- Not considering one-sided limits when checking if a limit exists at a point where the function definition changes.

5. Now Try It

Consider the function $g(x) = \frac{x^2 - 4}{x - 2}$.
Spend about 15 minutes determining if this function is continuous at $x=2$. Make sure to go through all three conditions for continuity step-by-step. What type of discontinuity is it, if any?

Success means you've correctly applied the three conditions, determined the function is not continuous at $x=2$, and correctly identified the type of discontinuity.