Topic 1.11 Notes – Defining Continuity at a Point

1. What It Means for a Function to Be Continuous at $x = c$

A function $f$ is continuous at $x = c$ if and only if all three of the following are true:

$$\text{1. } f(c) \text{ exists}$$

$$\text{2. } \lim_{x \to c} f(x) \text{ exists}$$

$$\text{3. } \lim_{x \to c} f(x) = f(c)$$

This is the definition. When you justify continuity on a quiz or FRQ, you are checking these three conditions explicitly.

Quick reminder about limits:

• A limit exists only if the left-hand limit and right-hand limit both exist and are equal.
• If they don’t match, the overall limit does not exist.

So continuity is really about this idea:
The function’s value matches what the function is approaching.

If even one condition fails, the function is not continuous at that point.

2. The Three Conditions Broken Down

Let’s slow down and look at what each condition actually means in practice.

a. $f(c)$ exists

You plug in $c$ into the function.

• If you get a real number → good.
• If it’s undefined (like division by zero) → continuity immediately fails.

Example idea:
If $f(x) = \frac{x^2 - 1}{x - 1}$, then at $x = 1$ the function is undefined unless it’s been redefined separately. That already tells you something is wrong with continuity.

Students often forget to check this and jump straight to the limit. Don’t skip it.

b. $\lim_{x \to c} f(x)$ exists

This means:

• $\lim_{x \to c^-} f(x)$ exists
• $\lim_{x \to c^+} f(x)$ exists
• They are equal

Situations where this fails:

• Jump discontinuity → left and right limits are different.
• Vertical asymptote → function goes to $\pm\infty$, so no finite limit.
• Wild oscillation → no single value being approached.

If the limit does not exist, you stop. The function is not continuous.

c. The limit equals the function value

Even if the first two conditions work, you still need:

$$\lim_{x \to c} f(x) = f(c)$$

This is where holes happen.

You might have:

• A limit that exists
• A defined function value
• But they are different numbers

That breaks condition 3.

3. What This Looks Like on a Graph

Here’s a visual of common situations at $x = c$:

Across these examples, focus on what is happening at the highlighted x-value in each panel.

When you read a graph:

• Filled dot at $x = c$ tells you $f(c)$.
• Watch what happens as you approach from left and right.
• Ask: do both sides head to the same y-value?
• Then check: does that value match the filled dot?

Always translate what you see into the three formal conditions.

4. How to Justify Continuity on an FRQ

If a question says “Justify your answer using the definition,” they are looking for structure.

A strong response usually follows this pattern:

1. State the value of $f(c)$.
2. Evaluate $\lim_{x \to c} f(x)$.
3. Compare them directly.
4. Conclude clearly:
   “Since $\lim_{x \to c} f(x) = f(c)$, the function is continuous at $x = c$.”
   or
   “Since $\lim_{x \to c} f(x) \neq f(c)$, the function is not continuous at $x = c$.”

If the limit doesn’t exist, say that explicitly and state which one-sided limits differ.

On AP free-response questions, missing one of the three conditions often costs the justification point even if your final conclusion is correct.