Removing Discontinuities

Historical Context & Motivation

The concept of removing discontinuities is intimately tied to the broader history of rigorizing the notion of continuity and limits in mathematical analysis. For centuries, mathematicians worked with functions intuitively, assuming that algebraic expressions behaved "nicely" everywhere they appeared. It was only when paradoxes and contradictions arose — particularly around infinite series, oscillating functions, and points where formulas broke down — that the need for precise definitions became apparent. The journey from informal manipulation to the modern epsilon-delta framework is one of the great intellectual achievements of nineteenth-century mathematics, and the ability to identify and repair removable discontinuities sits at its heart.

The central question this topic addresses is deceptively simple: if a function fails to be continuous at a single point, but the limit from both sides exists and agrees, can we "fix" the function? The answer is yes — and understanding exactly when and how to perform this repair is an essential skill in calculus. On the AP Calculus BC exam, you will encounter functions defined by algebraic expressions, piecewise rules, or rational forms that have isolated holes, and you must determine the value that restores continuity. This section of the course bridges the gap between evaluating limits and understanding what it truly means for a function to be continuous.

Core Principles & Definitions

Before we can remove a discontinuity, we need to be precise about what continuity requires and how it can fail. Recall that a function f is continuous at a point c if and only if three conditions hold simultaneously: f(c) is defined, the limit of f(x) as x approaches c exists, and that limit equals f(c). A removable discontinuity occurs when the two-sided limit exists but either f(c) is undefined or f(c) does not equal the limit. In contrast, a jump discontinuity arises when the left-hand and right-hand limits both exist but disagree, and an infinite (essential) discontinuity occurs when at least one side diverges to infinity. Only the first type — the removable discontinuity — can be repaired by redefining the function at a single point.

Visual Explanation

Anatomy of a Removable Discontinuity

In the diagram above, observe that the function f(x) = (x² − 4)/(x − 2) can be factored as (x − 2)(x + 2)/(x − 2), which simplifies to x + 2 for all x ≠ 2. The graph is therefore a straight line with slope 1 and y-intercept 2, except at x = 2 where the original expression is undefined (0/0). Both the left-hand limit and the right-hand limit as x → 2 equal 4, so the two-sided limit exists and equals 4. The open circle at (2, 4) visually represents the missing point — the removable discontinuity. By defining f(2) = 4, we fill in the hole and the function becomes the complete, unbroken line y = x + 2. This process of algebraic simplification followed by point-evaluation is the core technique you will use throughout this topic.

Mathematical Framework

The formal machinery behind removing discontinuities rests on the ε-δ definition of limits and the three-part definition of continuity. Here we present the key equations and logical structure that govern the process.

In practice, computing the limit L often requires one of several algebraic techniques: factoring and canceling common factors, multiplying by a conjugate to rationalize a numerator or denominator, applying trigonometric identities (such as the fundamental limit lim(θ→0) sin θ / θ = 1), or using L'Hôpital's Rule when the expression yields an indeterminate form 0/0 or ∞/∞. Once L is determined, the removed-discontinuity version of f is simply the piecewise function that equals the original expression away from c and equals L at c. It is worth noting that this redefined function and the original agree everywhere except at the single point x = c, so all derivative and integral computations away from c are unaffected.

Classifying Discontinuities

A critical skill on the AP Calculus BC exam is the ability to quickly classify a discontinuity as removable, jump, or infinite. The classification depends entirely on the behavior of the left-hand limit, the right-hand limit, and the function value at the point in question. The following diagram and table provide a systematic decision framework.

Worked Example

Removing a Discontinuity in a Rational Function

Consider the piecewise function f(x) = (x² − 5x + 6)/(x − 3) for x ≠ 3, and f(3) = k. Determine the value of k that makes f continuous at x = 3.

Techniques & Common Pitfalls

Removing discontinuities is not always as straightforward as factoring a quadratic. Many AP problems involve radical expressions, trigonometric functions, or exponential-logarithmic combinations that require different algebraic strategies. The table below summarizes the most common techniques and the contexts in which they appear.

Connections to Advanced Topics

Removing discontinuities is not merely an isolated limit-computation technique — it connects to several advanced topics that appear later in the AP Calculus BC curriculum and beyond. Understanding these connections deepens your appreciation for why continuity matters and how removable discontinuities appear in broader mathematical contexts.

Perhaps the most important connection for the AP BC exam is to the definition of the derivative itself. When you compute f′(a) = lim(h→0) [f(a + h) − f(a)]/h, the difference quotient is undefined at h = 0 (it yields 0/0). The derivative exists precisely when this removable discontinuity can be "filled in" — that is, when the limit exists. In this sense, every derivative computation is an exercise in removing a discontinuity from the difference quotient. This perspective also clarifies why differentiability implies continuity: if the 0/0 in the difference quotient can be resolved, the original function must have been continuous at that point.

Practice Problems

Summary