Continuity

Functions Without Breaks

A function is continuous at a point $a$ when its value and its limit agree there — no jump, no hole, no gap. Formally, three conditions must all hold:

1. $f(a)$ is defined,
2. $\lim_{x \to a} f(x)$ exists, and
3. they are equal: $\lim_{x \to a} f(x) = f(a)$.

Intuitively, you can draw a continuous function through $a$ without lifting your pen. A function continuous at every point of an interval is continuous on that interval. Polynomials, $\sin$, $\cos$, and exponentials are continuous everywhere; sums, products, and compositions of continuous functions are continuous.

When continuity fails, the discontinuity has a type:

• Removable — a single hole; the limit exists but $f(a)$ is missing or misplaced (fixable by redefining one point).
• Jump — the one-sided limits exist but differ; the graph leaps.
• Infinite — the function blows up to $\pm\infty$ (a vertical asymptote).

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

• The Intermediate Value Theorem: a continuous function takes every value between $f(a)$ and $f(b)$. So if $f$ changes sign on $[a, b]$, it must have a root in between — the basis of the bisection method for solving equations.
• The Extreme Value Theorem: a continuous function on a closed, bounded interval attains a maximum and a minimum — guaranteeing that optimization problems have solutions.

These theorems fail without continuity, which is why continuity is a prerequisite in so much of analysis.

Common pitfall: thinking a function is continuous merely because its limit exists at a point. Continuity requires all three conditions — the value $f(a)$ must exist and equal the limit. A removable discontinuity (a hole) has a perfectly good limit yet is not continuous, because $f(a)$ is missing or does not match. Existence of the limit is necessary but not sufficient.