Understanding Continuity (Calculus)

Note for review: this video's audio and visuals are identical to "Continuity (When the Limit Equals the Function Value)" (Tqy_pf-CJwU) — the same lesson appears to have been uploaded twice under different titles. Only one of the two should be published.

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

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 third condition is the point of the lesson: the limit must equal the actual function value.

Worked example

Determine whether $f(x) = x^2 + 1$ is continuous at $x = 2$.

Condition 1: $f(2) = 2^2 + 1 = 5$, so it is defined.

Condition 2: polynomials are continuous everywhere, so direct substitution is valid and $\displaystyle\lim_{x \to 2} f(x) = 2^2 + 1 = 5$.

Condition 3: both equal $5$, so they match.

All three hold, therefore $f$ is continuous at $x = 2$.

Exam note

On free-response questions, write the notation out fully and show each condition — marks are easily lost on presentation.