Continuity describes when a graph has no break, hole, jump, or vertical asymptote at a point. Students need this cheat sheet because many calculus topics, including derivatives and the Intermediate Value Theorem, depend on recognizing continuous behavior. It helps organize the three-part test for continuity and the major types of discontinuity in one clear reference.
A function is continuous at when is defined, exists, and . One-sided limits compare behavior from the left and right, using and . Discontinuities are commonly classified as removable, jump, infinite, or oscillating depending on what happens to the limit and function value.
Key Facts
- A function is continuous at if is defined, exists, and .
- The two-sided limit exists only when .
- A removable discontinuity occurs when exists but is undefined or .
- A jump discontinuity occurs when and both exist but are not equal.
- An infinite discontinuity occurs when at least one one-sided limit is infinite, such as or .
- A rational function may have a removable discontinuity where a common factor cancels, such as for .
- Polynomial functions are continuous for all real numbers, and rational functions are continuous wherever their denominators are not zero.
- Piecewise functions are checked for continuity at boundary points by comparing , , and .
Vocabulary
- Continuity
- A function is continuous at when the graph has no break there and .
- One-sided limit
- A one-sided limit describes the value a function approaches from only one side, written or .
- Removable discontinuity
- A removable discontinuity is a hole or misplaced point where exists but does not match the function value.
- Jump discontinuity
- A jump discontinuity occurs when the left-hand and right-hand limits at exist but have different values.
- Infinite discontinuity
- An infinite discontinuity occurs when a function grows without bound near , often because of a vertical asymptote.
- Oscillating discontinuity
- An oscillating discontinuity occurs when function values keep fluctuating near so no single limiting value exists.
Common Mistakes to Avoid
- Saying a function is continuous because is defined is wrong because continuity also requires to exist and equal .
- Canceling a factor and forgetting the original restriction is wrong because still has a hole at even though it simplifies to for .
- Assuming a two-sided limit exists when only one side has been checked is wrong because and must be equal.
- Calling every undefined point a vertical asymptote is wrong because an undefined point can be a removable discontinuity if the limit is finite.
- Ignoring piecewise boundary values is wrong because continuity at a boundary depends on the left rule, the right rule, and the actual value .
Practice Questions
- 1 Determine whether is continuous at , and classify any discontinuity.
- 2 For , find , , and decide whether is continuous at .
- 3 Find the value of that makes continuous at .
- 4 Explain why a graph with a hole at can have exist even though the function is not continuous at .
Understanding Continuity & Types of Discontinuity
A useful way to study continuity is to separate the graph’s nearby behavior from its actual plotted point. A graph can approach one height smoothly while the point at that input is missing or placed somewhere else. That distinction explains why a small open circle can matter so much.
In a table, use inputs increasingly close to the target from both sides. In a graph, trace the left branch and right branch toward the same vertical line.
In an equation, simplify carefully, but keep track of values excluded by the original expression. Simplifying can reveal the behavior near a point, yet it does not automatically restore a value that the original rule did not allow.
Removable discontinuities often arise from algebraic cancellation. For example, the expression x squared minus nine divided by x minus three simplifies to x plus three when x is not three. The simplified rule predicts nearby outputs, which approach six as x approaches three.
Still, the original denominator is zero at three. This creates a hole rather than a valid point. A function can be repaired at such a hole by defining its value to be the limiting height.
This idea is important because calculus often studies local behavior. A hole may prevent continuity at one input, even when values immediately around it follow a very simple pattern.
Not every break can be repaired. A jump represents an abrupt change between two separate output levels. Step models can have jumps, such as parking fees that change once a time threshold is passed or tax rules based on income brackets.
At the boundary, approaching from below describes one rule and approaching from above describes another. Infinite discontinuities occur near values that make a denominator zero without cancellation. The outputs can grow without bound in size, creating a vertical asymptote.
Oscillating discontinuities are different. The outputs keep switching or wavering with no single height to approach.
A classic example involves sine of one divided by x near zero. The values stay between negative one and one, but they never settle.
Piecewise functions require especially careful reading. First identify every boundary where the formula changes. Then evaluate the expression used on each side, rather than substituting into only one piece.
Finally inspect which piece includes the boundary itself. Brackets and inequality signs determine the actual function value. This habit prevents a common error where students find matching side behavior but overlook a dot placed at a different height.
Continuity later supports powerful conclusions. On an unbroken interval, a function cannot move from a negative output to a positive output without taking the value zero somewhere in between.
Derivatives likewise require continuity at the point of differentiation, though continuity alone does not guarantee a derivative. Corners, cusps, and vertical tangents can be continuous while still having no ordinary derivative.