Discontinuities | Tony's Intertwingled Memex

Core Definition

A function $f(x)$ is discontinuous at a point $x = c$ if it violates any of the three mandatory criteria for mathematical continuity:

$f(c)$ is defined (i.e., $c$ is in the domain of $f$ ).
$\lim_{x \to c} f(x)$ exists (as a finite real number $L$ ).
$\lim_{x \to c} f(x) = f(c)$ . If any condition fails, $x = c$ is classified as a point of discontinuity.

The Three Core Types of Discontinuities

Discontinuities are diagnosed and classified based on the behavioral failure of the limit and the function value:

                          Point of Discontinuity at x = c
                                         |
            +----------------------------+----------------------------+
            |                                                         |
Does the limit lim_{x->c} f(x) exist?                      Do the one-sided limits blow up
            |                                              to +/- infinity?
            +----------------+                                        |
            |                |                                        v
           YES               NO                             <a class="text-accent underline font-semibold" href="/garden/infinite-discontinuity">Infinite Discontinuity</a>
            |                |                              (Vertical Asymptote)
   Is f(c) != lim f(x)?      |                              e.g., f(x) = 1/(x-c)
            |                v
            v       Are one-sided limits
<a class="text-accent underline font-semibold" href="/garden/removable-discontinuity">Removable Discontinuity</a> finite but unequal?
(Hole in the graph)          |
e.g., f(x) = (x^2-1)/(x-1)   +------------> YES: <a class="text-accent underline font-semibold" href="/garden/jump-discontinuity">Jump Discontinuity</a>
                                                 (Step in the graph)
                                                 e.g., f(x) = |x|/x

1. Removable Discontinuity (Hole)

Diagnosis: $\lim_{x \to c} f(x) = L$ exists as a finite number, but either $f(c)$ is undefined or $f(c) \neq L$ .
Geometric Appearance: A smooth continuous curve with an isolated single-point hole (and possibly an isolated point plotted elsewhere at $x = c$ ).
Algebraic Cause: Arises from shared factors in rational functions that evaluate to the indeterminate form $\frac{0}{0}$ .
Resolution: "Removable" because redefining $f(c) = L$ completely repairs the discontinuity.

2. Jump Discontinuity (Step)

Diagnosis: The left-hand limit $\lim_{x \to c^-} f(x) = L_1$ and right-hand limit $\lim_{x \to c^+} f(x) = L_2$ both exist as finite numbers, but $L_1 \neq L_2$ .
Geometric Appearance: The curve abruptly breaks and steps vertically by a finite jump distance $\Delta y = |L_2 - L_1|$ .
Algebraic Cause: Piecewise-defined functions with mismatched boundary formulas or absolute value quotients like $\frac{|x-c|}{x-c}$ .
Resolution: Cannot be repaired by redefining a single point.

3. Infinite Discontinuity (Vertical Asymptote)

Diagnosis: One or both one-sided limits grow without bound ( $\lim_{x \to c^-} f(x) = \pm\infty$ or $\lim_{x \to c^+} f(x) = \pm\infty$ ).
Geometric Appearance: The graph shoots upward or downward towards infinity along a vertical asymptote line $x = c$ .
Algebraic Cause: Rational expressions evaluating to $\frac{k}{0}$ (where $k \neq 0$ ).

(Note: Functions that oscillate infinitely fast, such as $\sin(1/x)$ near $x = 0$ , represent an Essential / Wildly Oscillating Discontinuity).

Example Problems

Example Problem 1: Classifying Discontinuities Analytically

Classify the type of discontinuity at $x = 2$ for each function: (a) $f(x) = \frac{x^2 - 4}{x - 2}$
(b) $g(x) = \frac{1}{x - 2}$
(c) $h(x) = \begin{cases} x + 1 & : x < 2 \\ 2x - 3 & : x \ge 2 \end{cases}$

Solution

(a) $f(x) = \frac{x^2 - 4}{x - 2}$ : Direct substitution yields $\frac{0}{0}$ . Factoring gives $\lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2}(x+2) = 4$ . The two-sided limit exists, but $f(2)$ is undefined.
Classification: Removable Discontinuity (Hole at $(2, 4)$ ).
(b) $g(x) = \frac{1}{x - 2}$ : Direct substitution yields $\frac{1}{0}$ . As $x \to 2^-$ , $g(x) \to -\infty$ ; as $x \to 2^+$ , $g(x) \to +\infty$ .
Classification: Infinite Discontinuity (Vertical asymptote at $x = 2$ ).
(c) $h(x)$ : Left limit: $\lim_{x \to 2^-} (x+1) = 3$ . Right limit: $\lim_{x \to 2^+} (2x-3) = 1$ . Both limits are finite, but $3 \neq 1$ .
Classification: Jump Discontinuity (Jump magnitude $\Delta y = 2$ ).

Connections & References

Parent Concepts: Continuity, Limits
Sub-types: Removable Discontinuity, Jump Discontinuity, Infinite Discontinuity
Related Concepts: Existence Theorem for Two-Sided Limits, Piecewise Functions
Course Links: Calc-1 Session 02 | Calc-1 Session 03 | Calc-1 Session 05