Topic 1.11 Notes – Defining Continuity at a Point

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

Here is the formal definition you are expected to know and use:

A function $f$ is continuous at $x = c$ if and only if:

1. $f(c)$ exists
2. $\lim_{x \to c} f(x)$ exists
3. $\lim_{x \to c} f(x) = f(c)$

All three must be true. If even one fails, the function is not continuous at that point.

A quick reminder about limits:
A two-sided limit $\lim_{x \to c} f(x)$ exists only if the left-hand limit and right-hand limit both exist and are equal.

So continuity means the function’s value and its limiting behavior agree perfectly at $x = c$.

The Three Conditions Broken Down

Let’s slow this down and look at each requirement.

1. The Function Value Exists

You must be able to plug in $c$ and get a real number.

• If the function is undefined at $c$ (like dividing by zero), continuity already fails.
• On an FRQ, this is usually one short sentence: “$f(c)$ exists because …”

If there is no defined value, you are done. Not continuous.

2. The Limit Exists

This means:

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

Both one-sided limits must exist and match.

Ways this can fail:

• Jump discontinuity: left and right limits are different.
• Infinite discontinuity: function blows up to $\pm\infty$.
• Oscillation: function doesn’t settle to one value.

If the two sides disagree, the limit does not exist, so the function is not continuous.

3. The Limit Equals the Function Value

This is the condition students forget most often.

You might have:

• A defined function value
• A perfectly good limit

But if they are not equal, continuity fails.

That situation creates a removable discontinuity.

Seeing Continuity on a Graph

Here’s a visual comparison. Focus on the two removable-discontinuity panels across the top.

In the left top panel:

• The line approaches a value.
• The filled-in point is exactly at that value.
• Continuous at that $x$-value.

In the right top panel:

• The line approaches one value.
• The filled-in point is somewhere else.
• The limit exists, but it does not equal $f(c)$.
• Not continuous.

When you justify from a graph, say explicitly:

• The left-hand and right-hand limits are equal.
• The limit equals (or does not equal) the function value.

Avoid saying “it looks smooth.” That earns no credit.

How to Justify Continuity on an FRQ

When they say “justify using the definition,” they want structure.

A clean justification looks like this:

1. State whether $f(c)$ exists and why.
2. Compute or describe $\lim_{x \to c} f(x)$.
3. Compare the two.
4. Conclude clearly.

Example structure:

• “$f(c)$ exists because …”
• “$\lim_{x \to c} f(x) = …$”
• “Since the limit equals the function value, $f$ is continuous at $x = c$.”

If it fails, name the specific condition that fails. Graders look for that precision.

For piecewise functions, always:

• Check left-hand limit.
• Check right-hand limit.
• Evaluate the actual function value separately.

That breakpoint is almost always the whole question.

Common Types of Discontinuity at a Point

Tie each one back to the definition:

• Removable
  Limit exists, but either $f(c)$ is missing or not equal to the limit.
• Jump
  Left and right limits are not equal. Condition 2 fails.
• Infinite
  Function approaches infinity. The limit does not exist.

On tests, they often ask you to classify the discontinuity and justify it using which condition fails.