Exploring Types of Discontinuities

The modern concept of a continuous function seems intuitive — draw a curve without lifting your pen — yet making this idea precise required centuries of effort. Early mathematicians like Euler freely manipulated formulas and assumed functions behaved "nicely" between computed values, but as analysts studied Fourier series and pathological constructions in the 1800s, they realized that a rigorous definition of continuity was essential. Understanding where and how functions fail to be continuous became just as important as understanding continuity itself, because those failures — discontinuities — encode information about limits, domain restrictions, and the algebraic structure of a function.

The central question this lesson addresses is: when a function fails to be continuous at a point, what is the nature of that failure? On the AP Calculus BC exam, you must identify and classify discontinuities by analyzing one-sided limits, overall limits, and function values. The three classical types — removable, jump, and infinite (essential) — each arise from a different breakdown in the three-part definition of continuity, and recognizing them quickly is a foundational skill for everything from integration theory to series convergence.

A function f is continuous at x = c if and only if three conditions hold simultaneously: (1) f(c) is defined, (2) lim as x → c of f(x) exists, and (3) lim as x → c of f(x) = f(c). A discontinuity at x = c means at least one of these conditions fails. The specific condition that breaks determines the type of discontinuity.

In Figure 1, each panel isolates the signature visual fingerprint of its discontinuity type. For the removable case, the curve approaches the same y-value from both sides (the open circle), but f(c) is either undefined or sits at a different height (the filled dot). Algebraically, this usually arises from a common factor that cancels, such as (x² − 4)/(x − 2) at x = 2. For the jump case, the curve has two finite landing heights that simply do not match; piecewise-defined functions are the classic source. Finally, the infinite case exhibits the telltale vertical asymptote: at least one branch of the graph shoots toward ±∞ as x approaches c, which is the hallmark of a zero in the denominator that does not cancel.

A precise classification of discontinuities follows directly from the ε–δ definition of continuity. We say f is continuous at c when for every ε > 0 there exists δ > 0 such that |x − c| < δ implies |f(x) − f(c)| < ε. When this fails, we examine the one-sided limits and the function value to determine the discontinuity type.

When faced with a function and asked to classify a discontinuity, a systematic decision tree is far more reliable than intuition alone. The flowchart below encodes the logical hierarchy: first check whether one-sided limits are finite, then whether they agree, and finally whether the function value matches the limit.

When working through the flowchart, the most common algebraic strategy is to factor and simplify the expression. If a common factor cancels and the remaining expression is defined at c, the original discontinuity is removable. If no factor cancels and the denominator is zero, the discontinuity is infinite. For piecewise functions, compute both one-sided limits directly by substituting into the appropriate piece; if they differ, the discontinuity is a jump.

Let us classify every discontinuity of the function defined by:

Understanding discontinuities is not merely an exercise in classification — it has direct consequences throughout the AP Calculus BC curriculum and beyond. The Intermediate Value Theorem (IVT) requires continuity on a closed interval, so knowing where discontinuities lie tells you exactly where IVT conclusions are valid and where they may fail. Similarly, the Fundamental Theorem of Calculus requires the integrand to be continuous on [a, b]; when it is not, you must split the interval or invoke improper integral techniques.

In more advanced mathematics, the study of discontinuities connects to measure theory and Lebesgue integration, where the set of discontinuity points determines integrability (Lebesgue's criterion states that a bounded function is Riemann integrable if and only if its set of discontinuities has measure zero). While that level of formalism is beyond the BC exam, recognizing the foundational role of discontinuity classification motivates why we devote careful attention to it now.