AP Calculus AB and BC

Continuity vs Limit Existence

Q: Does continuity guarantee the limit exists?

Yes, that direction always holds. Continuity at a is defined to include the existence of the limit, so a continuous point never has a missing limit.

A limit existing is the weaker condition. The limit only asks what the function approaches from both sides, so it can exist at a hole where the function is undefined. Continuity asks for more: the limit must exist, f(a) must be defined, and those two numbers must be equal.

Continuity

Use when: The question asks whether the function is continuous at a point, or you need continuity as a hypothesis for the IVT or the EVT.

Limit existence

Use when: The question asks what value the function approaches, which you can answer even when the function has no value at that input.

Side by side

	Continuity	Limit existence
What it asks	Does the graph pass through the point with no break?	What one value does $f$ approach from both sides?
Condition	$\lim_{x \to a} f(x) = f(a)$	$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$ , both finite
Needs $f(a)$ to be defined	Yes	No
Holds at a removable hole	No	Yes
Common trap	Checking the limit and stopping there	Substituting $a$ to get $f(a)$ when the point is a hole

Continuity is a three-part test and the limit is only one part of it. The limit is a statement about the values of $f$ near $a$ , while continuity also requires the function to have a value at $a$ and requires that value to be the one the approach predicted.

$f(a)$ is defined
$\lim_{x \to a} f(x)$ exists
$\lim_{x \to a} f(x) = f(a)$

A removable hole is where the two part company. For $f(x) = \frac{x^2 - 9}{x - 3}$ the limit at $x = 3$ is $6$ , because the function equals $x + 3$ everywhere except at that one input, yet $f(3)$ does not exist. Step 2 passes, step 1 fails, so the limit exists and continuity does not.

Why the limit ignores the point itself

The definition of a limit describes $x$ near $a$ and deliberately excludes $x = a$ . Deleting or moving the single value $f(a)$ therefore cannot change the limit, which is exactly what makes a hole invisible to the limit and fatal to continuity.

Frequently asked questions

Can a limit exist where the function is undefined?

Yes. At a removable hole the two-sided limit exists even though $f(a)$ does not, which is why these are two separate questions.

Does continuity guarantee the limit exists?

Yes, that direction always holds. Continuity at $a$ is defined to include the existence of the limit, so a continuous point never has a missing limit.

How do I check continuity at a piecewise seam?

Take both one-sided limits, confirm they agree, then confirm that shared value equals the value the definition assigns at the seam. All three steps are needed.

In the CED: Unit 1: Limits and Continuity