Continuity At A Point

Continuity at a Point

A function is continuous at a point if its graph has no hole, jump, or break there—you could draw it without lifting your pencil. To test continuity rigorously, we verify three specific conditions rather than relying on intuition.

The Three Essential Conditions

A function $f$ is continuous at $x = a$ if and only if all three hold:

If any condition fails, $f$ is discontinuous at $a$.

Example: Testing Continuity

Consider $f(x) = \dfrac{x^2 - 4}{x - 2}$ for $x \neq 2$, with $f(2) = 5$.

• Condition 1: $f(2) = 5$ ✓
• Condition 2: Factor to get $\dfrac{(x-2)(x+2)}{x-2} = x+2$ for $x \neq 2$, so $\lim_{x\to 2} f(x) = 4$ ✓
• Condition 3: $4 \neq 5$ ✗

Result: discontinuous (removable type).

Types of Discontinuity

Removable discontinuity occurs when the limit exists but doesn't equal $f(a)$, or when $f(a)$ is undefined but the limit exists—redefining one value would "fix" it.

Jump discontinuity occurs when the left and right limits both exist but disagree, creating an abrupt step in the graph.

Infinite discontinuity occurs when at least one one-sided limit is infinite, producing a vertical asymptote.

Continuity on Intervals

On an open interval $(a,b)$, check continuity at every interior point. On a closed interval $[a,b]$, additionally require one-sided continuity at endpoints: $\lim_{x\to a^+} f(x) = f(a)$ and $\lim_{x\to b^-} f(x) = f(b)$.