Continuity of Piecewise Defined Functions

TL;DR

A piecewise function is continuous if each individual piece is continuous and if the pieces connect seamlessly where they meet. To check for continuity, you'll need to evaluate the function at the "break points" and ensure the limits match the function value. This process helps you understand how functions behave across different definitions.

1. The Mental Model

Imagine you're drawing a function without lifting your pencil. If your function changes its definition mid-draw (that's a piecewise function!), you want to make sure your pencil can smoothly transition from one piece to the next without any jumps, holes, or sudden breaks.

2. The Core Material

A function is continuous at a point if three conditions are met:
1. The function is defined at that point (no holes).
2. The limit of the function exists at that point (no jumps).
3. The function's value at the point equals its limit at that point (the value fills the gap).

For piecewise functions, we're primarily concerned with the points where the function's definition changes. These are called break points or transition points.

Checking for Continuity at a Break Point

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Let's say your function $f(x)$ changes its definition at $x=c$. To check if $f(x)$ is continuous at $x=c$, you need to do three things:

1. Check if $f(c)$ is defined: Find which piece of the function applies at $x=c$ and calculate $f(c)$. It must exist.
2. Check if $\lim_{x \to c} f(x)$ exists: This means the limit from the left must equal the limit from the right.
   • Left-hand limit: $\lim_{x \to c^-} f(x)$ (use the piece defined for $x < c$).
   • Right-hand limit: $\lim_{x \to c^+} f(x)$ (use the piece defined for $x > c$).
     If these two limits are not equal, then the overall limit doesn't exist, and the function is not continuous at $c$.
3. Check if $f(c) = \lim_{x \to c} f(x)$: If both the function value and the limit exist, they must be equal for the function to be continuous at $x=c$.

If any of these conditions fail, the function is discontinuous at that break point. Remember, you also need to ensure that each individual piece is continuous on its own defined interval. Often, polynomial and rational functions (where the denominator isn't zero) are continuous on their intervals.

Here's how to think about the flow:

graph TD
    A["Identify Break Points"] --> B{"Is f(c) Defined?"}
    B -- Yes --> C{"Does lim(x->c) f(x) exist?"}
    C -- Yes --> D{"Is f(c) = lim(x->c) f(x)?"}
    D -- Yes --> E["Continuous at x=c"]
    B -- No --> F["Discontinuous at x=c"]
    C -- No --> F
    D -- No --> F
    A --> G["Check each piece's continuity on its own interval"]
    G --> H["Combined Result"]

3. Worked Example

Let's examine the continuity of the function:
$$ f(x) = \begin{cases} x^2 + 1 & \text{if } x < 1 \\ 3x - 1 & \text{if } x \ge 1 \end{cases} $$

Our only break point is at $x=1$.

1. Is $f(1)$ defined?
   Since $x \ge 1$ for the second piece, we use $f(x) = 3x - 1$.
   $f(1) = 3(1) - 1 = 3 - 1 = 2$.
   Yes, $f(1) = 2$.

2. Does $\lim_{x \to 1} f(x)$ exist?

   • Left-hand limit: For $x < 1$, we use $f(x) = x^2 + 1$.
     $\lim_{x \to 1^-} (x^2 + 1) = (1)^2 + 1 = 1 + 1 = 2$.
   • Right-hand limit: For $x > 1$, we use $f(x) = 3x - 1$.
     $\lim_{x \to 1^+} (3x - 1) = 3(1) - 1 = 3 - 1 = 2$.
     Since the left-hand limit equals the right-hand limit (both are 2), $\lim_{x \to 1} f(x)$ exists and is equal to 2.

3. Is $f(1) = \lim_{x \to 1} f(x)$?
   We found $f(1) = 2$ and $\lim_{x \to 1} f(x) = 2$.
   Since $2 = 2$, this condition is met.

Since all three conditions are satisfied at $x=1$, and both individual pieces ($x^2+1$ and $3x-1$ are polynomials, hence continuous everywhere), the function $f(x)$ is continuous for all real numbers.

4. Key Takeaways

• Continuity means you can trace the function's graph without lifting your pen.
• For piecewise functions, the most important points to check are where the definition changes (break points).
• At a break point, the function value, the left-hand limit, and the right-hand limit must all exist and be equal.
• Always check that each piece of the function is continuous on its own specified interval.
• Polynomials, exponentials, sines, and cosines are always continuous on their domains. Rational functions are continuous everywhere their denominator isn't zero.
• If you're dealing with a parameter (like 'k') in a piecewise function, set the left and right limits equal at the break point to solve for the parameter value that makes it continuous.

Common Mistakes:
- Forgetting to check if $f(c)$ is actually defined at the break point.
- Only checking one side of the limit instead of both left-hand and right-hand limits.
- Not using the correct function piece for calculating $f(c)$, the left-hand limit, or the right-hand limit.
- Assuming a function is continuous just because its individual pieces are. The connection point is crucial!

5. Now Try It

Consider the function:
$$ g(x) = \begin{cases} 2x + 3 & \text{if } x \le 2 \\ x^2 + a & \text{if } x > 2 \end{cases} $$
Find the value of 'a' that makes $g(x)$ continuous everywhere. What success looks like: you'll have a single numerical value for 'a' that ensures the two pieces meet perfectly at $x=2$.