A function is continuous at a point when its graph can be drawn through that point without a break, hole, jump, or vertical blow-up. Discontinuities matter because they show where a formula changes behavior or where a model stops making physical or mathematical sense. In calculus, continuity is also a key condition for theorems about limits, derivatives, and integrals.
Learning to recognize the main types helps you read graphs and analyze piecewise functions correctly.
At a point x = a, continuity requires three things: f(a) is defined, lim x -> a f(x) exists, and the limit equals the function value. A removable discontinuity happens when the two-sided limit exists but the function value is missing or placed at the wrong height. A jump discontinuity happens when the left-hand and right-hand limits are finite but unequal.
An infinite discontinuity happens when function values grow without bound near x = a, often because of a vertical asymptote.
Understanding Calculus: Types of Discontinuity
The best way to classify a break is to inspect the function from both sides before looking at the plotted point itself. Imagine walking along the graph from values smaller than a chosen input, then from values larger than it. Record the height approached on each trip.
If both trips head toward one ordinary height, the break may be removable. If they settle at different heights, it is a jump.
If the heights rise or fall without limit, the function has an infinite discontinuity. This side by side approach prevents a common error where students use only one side of a graph.
Algebra can reveal the same behavior without a graph. For a rational expression, first factor the numerator and denominator completely. A shared factor may cancel for inputs near a value, but the original denominator still makes that value unavailable.
The remaining expression tells the height of the hole. For example, a numerator containing the factor x minus three and a denominator containing x minus three can simplify away from three.
If no copy of that denominator factor cancels, division by a number increasingly close to zero can produce extremely large positive or negative outputs. That pattern signals a vertical asymptote rather than a hole.
Piecewise rules are especially important for jump discontinuities. Each formula controls only part of the graph, so substitute the boundary input into the formula used on the left and the formula used on the right. The actual assigned value at the boundary can be a third number.
It does not repair a jump because changing one plotted dot cannot make two different approaching heights match. Step functions used in parking fees, postage costs, tax brackets, and digital measurement settings often have jumps.
They model a quantity that changes in fixed stages rather than smoothly. A graph can therefore be useful even when its discontinuity represents a deliberate rule.
Discontinuities affect which calculus tools are safe to use. A derivative requires local smooth behavior, so a function cannot have a derivative at a hole, jump, or vertical asymptote. A definite integral can still exist across some discontinuities.
A single removable hole does not change the accumulated area, and a finite number of jumps can often be handled by splitting the interval. An infinite discontinuity needs much more care because the area near the asymptote may be finite or may grow without bound. When solving a problem, mark excluded inputs early, check one sided behavior, simplify only with domain restrictions remembered, then state clearly what type of break occurs.
Key Facts
- Continuity at x = a requires f(a) is defined, lim x -> a f(x) exists, and lim x -> a f(x) = f(a).
- Two-sided limit exists only when lim x -> a- f(x) = lim x -> a+ f(x).
- Removable discontinuity: lim x -> a f(x) exists, but f(a) is undefined or f(a) != lim x -> a f(x).
- Jump discontinuity: lim x -> a- f(x) and lim x -> a+ f(x) are finite but not equal.
- Infinite discontinuity: f(x) -> infinity or f(x) -> -infinity as x approaches a from at least one side.
- A rational function often has a removable discontinuity when a factor cancels and an infinite discontinuity when a noncanceling denominator factor equals zero.
Vocabulary
- Continuity
- Continuity at a point means the function value matches the limit of the function as x approaches that point.
- Removable discontinuity
- A removable discontinuity is a hole or misplaced point where the limit exists but the function is not defined correctly at that x-value.
- Jump discontinuity
- A jump discontinuity occurs when the left-hand and right-hand limits are different finite numbers.
- Infinite discontinuity
- An infinite discontinuity occurs when the function grows without bound near a point, usually creating a vertical asymptote.
- One-sided limit
- A one-sided limit describes the value a function approaches from only the left side or only the right side of a point.
Common Mistakes to Avoid
- Ignoring whether f(a) is defined. A limit can exist at x = a even if the function has a hole there, so you must check the actual function value separately.
- Assuming a graph is continuous because both sides look close. For continuity, the left-hand limit, right-hand limit, and f(a) must all be the same number.
- Calling every break a vertical asymptote. A hole is removable and a jump has finite one-sided limits, while an infinite discontinuity has values that grow without bound.
- Canceling factors and forgetting the original restriction. If (x - 2) cancels from a rational expression, x = 2 may still be excluded from the original function and can create a removable discontinuity.
Practice Questions
- 1 For f(x) = (x^2 - 9)/(x - 3), x != 3, identify the type of discontinuity at x = 3 and find lim x -> 3 f(x).
- 2 Let f(x) = 2x + 1 for x < 1 and f(x) = x^2 + 4 for x >= 1. Find lim x -> 1- f(x), lim x -> 1+ f(x), and state the type of discontinuity at x = 1.
- 3 A graph has an open circle at (2, 5), a filled dot at (2, 1), and the curve approaches y = 5 from both sides. Explain whether the function is continuous at x = 2 and name the type of discontinuity.