1.10 Exploring Types of Discontinuities

1.10 Exploring Types of Discontinuities

Limits describe where a function is heading near an input; this page uses them to describe what goes wrong at an input where the graph breaks. One example will be reused: the sensor model $T(x)=\frac{x^2-9}{x-3}$ from Topic 1.9 is undefined at $x=3$ but equals $x+3$ for every other $x$, so $\lim_{x\to3}T(x)=6$.

Reading discontinuities as mismatches

A discontinuity occurs where a function’s local behavior fails to fit together smoothly. To explore the type of discontinuity at $x=a$, compare three pieces of information:

$$ \lim_{x\to a^-}f(x),\qquad \lim_{x\to a^+}f(x),\qquad f(a). $$

The representations reveal different failure patterns:

Type	What the graph shows	Limit behavior
Removable discontinuity	A hole, possibly with a separately plotted point	The two-sided limit exists, but $f(a)$ is missing or differs
Jump discontinuity	Two branches meet different heights	The one-sided limits exist but are unequal
Infinite discontinuity	The graph rises or falls without bound near $x=a$	At least one one-sided limit is $\infty$ or $-\infty$

For the sensor model above, the discontinuity at $x=3$ is removable: both sides approach $6$, but the original expression has no value at $x=3$. Defining a new function with $T(3)=6$ would fill the hole without changing the surrounding limit.

By contrast, consider a water-flow display defined by

$$ R(t)= \begin{cases} 4, & t<2,\ 7, & t\ge 2. \end{cases} $$

Then

$$ \lim_{t\to 2^-}R(t)=4 \quad\text{and}\quad \lim_{t\to 2^+}R(t)=7. $$

Because the one-sided limits disagree, $\lim_{t\to 2}R(t)$ does not exist. This is a jump discontinuity, regardless of the value assigned at $t=2$.

For

$$ P(x)=\frac{1}{(x-5)^2}, $$

the function grows without bound from both sides as $x$ approaches $5$:

$$ \lim_{x\to 5^-}P(x)=\infty \quad\text{and}\quad \lim_{x\to 5^+}P(x)=\infty. $$

This is an infinite discontinuity, associated with a vertical asymptote at $x=5$.

Misconception check: “If $f(a)$ is undefined, the limit does not exist.”
Correction: A function may be undefined at $a$ while its two-sided limit exists. A hole is precisely the graphical signature of this possibility.

Retrieval check

A graph has a hole at $(2,5)$, and the curve approaches $5$ from both sides. A filled point appears at $(2,8)$. State $\lim_{x\to 2}f(x)$, $f(2)$, and the type of discontinuity.

The answers are

$$ \lim_{x\to 2}f(x)=5,\qquad f(2)=8, $$

and the discontinuity is removable. The limit follows the approaching curve; the function value follows the filled point.