Removing Discontinuities
A function is continuous at a point x = a if three conditions hold: the function f(a) is defined, the limit lim_{x→a} f(x) exists, and the limit equals the function value. When a function fails one or more of these conditions at an isolated point, it has a removable discontinuity — a single 'hole' in the graph where the function is either undefined or assigned the wrong value. The good news is that such gaps can be repaired by redefining or extending the function at exactly that one point. Removing a discontinuity means choosing a specific output value at the problem point so that the repaired function becomes continuous there. The key is that the limit of the function as x approaches the point must already exist; if it does, we simply set the function's value equal to that limit. This technique appears constantly in calculus — most importantly when evaluating derivatives, since the definition of the derivative itself involves a limit of a difference quotient that is initially 0/0 at the point of interest. In this article you will learn to identify removable discontinuities, compute the limiting value that fills the hole, and write the piecewise-defined extension that restores continuity. You will also see why not every discontinuity is removable: jump discontinuities and infinite discontinuities cannot be fixed by reassigning a single value, because the two-sided limit either does not exist or diverges.
- Identify the problem point
- Check that the two-sided limit exists
- Simplify algebraically to find the limit L
- Define or redefine the function at x = a
- Verify continuity at the point
Identify the problem point
Find the value x = a where the function is undefined or discontinuous. Common causes are a zero denominator, a 0/0 indeterminate form after substitution, or a piecewise definition with mismatched pieces. For example, f(x) = (x² − 4)/(x − 2) is undefined at x = 2 because both numerator and denominator equal zero there.
Check that the two-sided limit exists
Compute lim_{x→a} f(x) from both the left and right. If the one-sided limits disagree, the discontinuity is a jump and cannot be removed. If the limit is ±∞, the discontinuity is infinite and also cannot be removed. Only when lim_{x→a⁻} f(x) = lim_{x→a⁺} f(x) = L (a finite number) do we have a removable discontinuity.
Simplify algebraically to find the limit L
Factor, rationalise, or apply trigonometric identities to cancel the problematic factor. For f(x) = (x² − 4)/(x − 2), factor the numerator: (x − 2)(x + 2)/(x − 2). For x ≠ 2 this equals x + 2, so lim_{x→2} f(x) = 2 + 2 = 4. The hole sits at the point (2, 4).
Define or redefine the function at x = a
Create an extended function g(x) that agrees with f(x) everywhere except at a, and set g(a) = L. Written as a piecewise function: g(x) = f(x) for x ≠ a, and g(a) = L. Now all three continuity conditions are satisfied at x = a: g(a) = L is defined, the limit equals L, and they agree. The hole is filled.
Verify continuity at the point
Confirm lim_{x→a} g(x) = g(a) = L. Because g(x) = f(x) for x ≠ a, the limit is unchanged, and by construction g(a) = L. The function g is continuous at x = a. You should also check that g remains continuous on the rest of its domain — the repair at one point does not affect behaviour elsewhere.
The Key Equation
- g(x) — Extended (repaired) function (dimensionless): The new continuous function obtained after filling the hole.
- f(x) — Original function (dimensionless): The function that has a removable discontinuity at x = a.
- a — Problem point (dimensionless): The x-value where f is undefined or has the wrong value.
- L — Limit value (dimensionless): The finite value that both one-sided limits approach; the value assigned to g(a).
Worked Example
Problem. Define the value of the constant c so that h(x) is continuous everywhere, where h(x) = (x² + x − 6)/(x − 2) for x ≠ 2, and h(2) = c.
Answer. c = 5
Origins · 1 minAugustin-Louis Cauchy and how this was discovered
Cauchy was the first mathematician to give a rigorous, precise definition of continuity and limits, publishing his foundational Cours d'analyse in 1821. His epsilon-delta framework replaced vague geometric intuitions with exact algebraic conditions, making it possible to state clearly what it means for a limit to exist or a function to be continuous at a point — the very machinery used to define and remove discontinuities. Cauchy was extraordinarily prolific, producing nearly 800 research papers. The French Academy of Sciences had to introduce a page limit on submissions largely because of him.
Related Concepts
- Limit of a Function — Removing a discontinuity depends entirely on the two-sided limit existing and being finite; understanding limits is prerequisite knowledge.
- Definition of Continuity — The three-part definition of continuity (function defined, limit exists, they agree) is the precise standard a repaired function must satisfy.
- Indeterminate Forms (0/0) — Most removable discontinuities arise from 0/0 indeterminate forms that require algebraic simplification to resolve.
- The Derivative as a Limit — The derivative is defined by a difference quotient that has a removable discontinuity at h = 0; removing it is exactly how differentiation works.