Continuity (When the Limit Equals the Function Value)

The informal definition

A function is continuous if you can draw its graph without lifting your pencil — no holes, no jumps, no breaks.

The formal definition

That plain-language version is not enough for the exam. Formally, $f$ is continuous at $x = a$ if and only if all three conditions hold:

1. $f(a)$ is defined
2. $\displaystyle\lim_{x \to a} f(x)$ exists
3. $\displaystyle\lim_{x \to a} f(x) = f(a)$

The video stresses the third condition as the whole point: the limit has to equal the actual function value at that point.

Worked example

Determine whether $f(x) = x^2 + 1$ is continuous at $x = 2$. (This is the graph of $x^2$ shifted up one unit.)

Condition 1 — is $f(2)$ defined? Substitute:

$f(2) = 2^2 + 1 = 5 \quad \checkmark$

Condition 2 — does the limit exist? This is a polynomial, and polynomials are continuous everywhere, so direct substitution is valid:

$\lim_{x \to 2} f(x) = 2^2 + 1 = 5 \quad \checkmark$

Condition 3 — do they match? Both give $5$, so yes.

All three conditions hold, therefore $f$ is continuous at $x = 2$ — and that is exactly why the limit and the function value are the same number.

Exam note

On a free-response question, write the notation out properly and show each condition. Marks are easily lost on sloppy justification even when the answer is right.