1.10 Exploring Types of Discontinuities

Learning objectives

—LIM-2.A—Justify conclusions about continuity at a point using the definition• LIM-2.A Justify conclusions about continuity at a point using the definition.• LIM-2.A.1 Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes.• Enduring understanding LIM-2: Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

Classify a discontinuity by comparing the function's behavior as $x$ approaches the point from the left and right, then checking whether the point value matches that nearby behavior.

Type	Nearby limit behavior at $x=c$	Typical graph feature
Removable	Both sides approach the same finite $L$ , but $f(c)$ is missing or $f(c)\ne L$	A hole, possibly with a filled point at another height
Jump	The finite left-hand and right-hand limits exist but are unequal	Two branches approach different heights
Vertical asymptote	At least one side is unbounded as $x\to c$	Values grow without bound near the vertical line $x=c$

$\dfrac{x^2-1}{x-1}$ is undefined at $x=1$ but approaches $2$ : removable.
If the left side approaches $2$ and the right side approaches $5$ : jump.
$\dfrac{1}{x-2}$ becomes unbounded near $x=2$ : vertical-asymptote discontinuity.

A useful order is: find the two one-sided limits; decide whether a finite two-sided limit exists; then compare it with $f(c)$ only when that finite limit exists. This separates a removable point-value mismatch from a jump or unbounded failure of the two-sided limit.

A hole does not automatically make the limit nonexistent: nearby outputs may still approach one finite value. Conversely, $\infty$ is not a finite function value at a vertical asymptote; it describes unbounded behavior. The full three-condition definition of continuity is developed in Topic 1.11.

Next Topic

—1.11 Defining Continuity at a PointRead