Continuity of a Function
Understanding when and why functions behave smoothly — and what happens when they don't
1. Definition of Continuity at a Point
The Three-Condition Checklist
A function $f$ is continuous at a point $x = a$ if and only if all three conditions hold simultaneously:
- $f(a)$ is defined — the point $a$ is in the domain of $f$.
- $\displaystyle\lim_{x \to a} f(x)$ exists — the function approaches a single finite value as $x$ approaches $a$.
- $\displaystyle\lim_{x \to a} f(x) = f(a)$ — the limit value equals the function value.
If any one of these conditions fails, the function is discontinuous at $a$.
Increment (Informal) Form
Continuity can be restated intuitively: a small change in the input produces a small change in the output. If we write $\Delta x = x - a$ and $\Delta f = f(a + \Delta x) - f(a)$, then continuity at $a$ means:
$$ \Delta x \to 0 \implies \Delta f \to 0 $$Epsilon–Delta (Rigorous) Form
The formal definition, first articulated by Bolzano and later refined by Weierstrass, states:
Reading it: No matter how tight a tolerance $\varepsilon$ you demand on the output, you can always find a tolerance $\delta$ on the input that guarantees it.
One-Sided Continuity
We say $f$ is right-continuous at $a$ if $\displaystyle\lim_{x \to a^+} f(x) = f(a)$, and left-continuous at $a$ if $\displaystyle\lim_{x \to a^-} f(x) = f(a)$.
Continuity on an Interval
- Open interval $(a, b)$: $f$ is continuous at every point in $(a, b)$.
- Closed interval $[a, b]$: $f$ is continuous on $(a, b)$, plus right-continuous at $a$ and left-continuous at $b$.
The endpoint convention is essential — at a boundary, only the one-sided limit toward the interior is required.