Continuity

Defining Continuity

A function is continuous at a point $a$ when you can draw it through $a$ without lifting your pen. Formally, three conditions must all hold simultaneously:

Condition	Requirement
1. Value	$f(a)$ is defined
2. Limit	$\lim_{x \to a} f(x)$ exists
3. Agreement	$\lim_{x \to a} f(x) = f(a)$

Polynomials, $\sin$ , $\cos$ , and exponentials are continuous everywhere, as are sums, products, and compositions of continuous functions.

Common pitfall: Thinking a function is continuous merely because its limit exists at a point. A removable discontinuity (a hole) has a perfectly good limit yet is not continuous, because $f(a)$ is missing or mismatched. Existence of the limit is necessary but not sufficient.

Two "yes" checks and one "no" -- and the pen still has to jump

Types and Theorems

When continuity fails, the discontinuity falls into three classic types:

Type	Description
Removable	A single hole; limit exists but $f(a)$ is missing or misplaced.
Jump	One-sided limits exist but differ; the graph leaps.
Infinite	The function blows up to $\pm\infty$ at a vertical asymptote.

Continuity powers two cornerstone theorems on a closed interval $[a, b]$ :

The Intermediate Value Theorem: Takes every value between $f(a)$ and $f(b)$ . If $f$ changes sign, it has a root in between (basis of bisection).
The Extreme Value Theorem: Attains a maximum and minimum, guaranteeing optimization solutions exist.

These theorems fail entirely without continuity.

Defining Continuity

Types and Theorems

Practise this lesson

Differential Calculus