Continuity means a function has no holes, jumps, or vertical asymptotes at a point: the limit exists, the function value exists, and they match. Different function families (polynomials, rationals, roots, trig composites) each follow predictable rules for where they are continuous, and stating those intervals in proper notation is the core skill tested here.
Difficulty: Introductory
Prerequisites: Limits (Section 1.3 – 1.6), basic algebra (factoring, domain restrictions). You should be comfortable evaluating one-sided limits and know what interval notation looks like.
Continuity is the bridge between limits (which you have just finished) and derivatives (which come next). A function that is continuous at a point behaves predictably there: you can evaluate the limit simply by plugging in. Nearly every theorem you will meet in Calculus I, from the Intermediate Value Theorem to the Extreme Value Theorem, requires continuity as a hypothesis, so knowing exactly where a function is and is not continuous is foundational. This assignment drills the two sides of that coin: writing the domain of continuity in interval notation, and classifying what goes wrong at points where continuity fails.
Continuous at a point
A function f is continuous at x = a when three conditions all hold: f(a) is defined, lim(x→a) f(x) exists, and lim(x→a) f(x) = f(a). Think of it as: you can draw through that point without lifting your pen, and the function value is exactly where the limit says it should be.
Interval of continuity
The set of all x-values where a function is continuous, written in interval notation. In simple terms, it is the answer to "where can I trust this function to behave nicely?"
Removable discontinuity
A point where the limit exists but either f(a) is not defined or f(a) does not equal the limit. Think of it as a single hole in the graph: the curve is fine on both sides, but someone punched out exactly one point (or stuck it in the wrong place).
Jump discontinuity
A point where the left-hand and right-hand limits both exist but are not equal. In simple terms, the graph makes a sudden step, like a staircase.
Infinite discontinuity
A point where at least one one-sided limit is ±∞, typically a vertical asymptote. Think of it as the graph blowing up toward infinity on one or both sides of that x-value.
Domain
The set of all input values for which the function is defined. This is the starting point for finding intervals of continuity: a function can only be continuous where it exists in the first place.
Interval notation
A compact way to describe sets of real numbers using parentheses (open, excluding the endpoint) and brackets (closed, including the endpoint). For example, [7/9, ∞) means all x from 7/9 onward, including 7/9 itself.
Every polynomial is continuous on (−∞, ∞).
This includes constants, linear functions, quadratics, and any sum of terms like 6x⁸ − 3x² + 3.
No domain restrictions at all, so the interval of continuity is always (−∞, ∞).
Why: polynomials are built from addition, subtraction, and multiplication of continuous pieces (x and constants), and sums/products of continuous functions are continuous.
A rational function p(x)/q(x) is continuous everywhere except where q(x) = 0.
To find the intervals of continuity, factor the denominator, find its zeros, and exclude them.
Example: f(x) = (x² − 9)/(x² − 8x + 15). Factor the denominator: x² − 8x + 15 = (x − 3)(x − 5). The zeros are x = 3 and x = 5.
The interval of continuity is (−∞, 3) ∪ (3, 5) ∪ (5, ∞).
Always use union (∪) to stitch together the pieces around excluded points.
f(x) = √(expression) is continuous wherever the expression under the root is ≥ 0.
Solve the inequality to find the domain, and that domain is the interval of continuity.
Example: f(x) = √(9x − 7). Solve 9x − 7 ≥ 0, giving x ≥ 7/9. The interval of continuity is [7/9, ∞).
The bracket is closed at 7/9 because the function is defined (equals zero) at that endpoint.
f(x) = ∛(expression) is continuous on (−∞, ∞), regardless of what sits under the root.
Example: f(x) = ∛(8x − 3) is continuous on (−∞, ∞).
Why: cube roots accept negative inputs, unlike even roots. There is no domain restriction.
sin(x) and cos(x) are continuous on (−∞, ∞).
A composition like sin(g(x)) is continuous wherever g(x) is continuous.
Example: lim(x→π) sin(3x − sin(6x)). Since 3x − sin(6x) is continuous everywhere (it is a difference of a polynomial and a sine), we evaluate by direct substitution: sin(3π − sin(6π)) = sin(3π − 0) = sin(3π) = 0.
Whenever a function is continuous at a point, the limit equals the function value. That is the whole power of continuity: direct substitution works.
\text{f is continuous at } x = a \iff \lim_{x \to a} f(x) = f(a)\text{Domain of } \sqrt{g(x)}: \quad g(x) \geq 0\text{Domain of } \frac{p(x)}{q(x)}: \quad q(x) \neq 0Cube roots have no domain restriction: ∛(g(x)) is defined for all real g(x).
Continuity matters whenever a model needs to behave predictably. The temperature of a cup of coffee cooling over time is a continuous function: it does not teleport from 90°C to 20°C. Engineers rely on continuous models to design control systems, because a sudden jump in a signal can cause hardware to malfunction. Whenever you hear "no sudden jumps," you are hearing a continuity assumption.
Students often think "continuous" means "has no breaks in its formula." A piecewise function can be perfectly continuous if the pieces meet at the boundary. Continuity is about the graph, not the number of formulas.
Students often confuse "not in the domain" with "discontinuous." If a function is not defined at a point, that point is simply not in its domain. Discontinuity only applies at points where we could reasonably ask whether the function is continuous (i.e. at or near points in its domain).
Students sometimes write (−∞, 3) ∪ (5, ∞) for the rational function example and forget the middle interval (3, 5). Every gap between excluded points is its own interval.
Students often assume cube roots behave like square roots. They do not. Cube roots handle negative inputs, so ∛(negative number) is perfectly real and defined.
⚠️ Interval notation must be exact. Parentheses vs. brackets matters: use [ ] only when the endpoint is included in the domain. A square root at its boundary gets a bracket; points excluded from a rational function get parentheses.
⚠️ Know the three conditions for continuity cold. Exam questions will test each one separately: "the limit exists but does not equal f(a)" (removable), "the one-sided limits differ" (jump), "a one-sided limit is infinite" (infinite).
⚠️ Direct substitution for limits of continuous functions is heavily tested. If you can show the function is continuous at the point, the limit is just the function value. This is the payoff of this entire section.
⚠️ Classifying discontinuity type from a graph or a piecewise formula is a near-certain exam item.
True or false: Every polynomial is continuous on (−∞, ∞). _____
Fill in the blank: f(x) = √(9x − 7) is continuous on the interval _____.
True or false: f(x) = ∛(8x − 3) has a restricted domain. _____
Fill in the blank: For f(x) = (x² − 9)/(x² − 8x + 15), the denominator equals zero at x = _____ and x = _____.
True or false: If lim(x→a) f(x) exists but f(a) is defined as a different value, the discontinuity is a jump. _____
Answers: 1. True. 2. [7/9, ∞). 3. False. 4. x = 3 and x = 5. 5. False (it is removable).
Q: Where is f(x) = 6x⁸ − 3x² + 3 continuous?
A: (−∞, ∞). It is a polynomial, and polynomials are continuous everywhere.
Q: Where is f(x) = (x² − 9)/(x² − 8x + 15) continuous?
A: Factor the denominator: (x − 3)(x − 5). Excluded points are x = 3 and x = 5. The interval of continuity is (−∞, 3) ∪ (3, 5) ∪ (5, ∞).
Q: Where is f(x) = √(9x − 7) continuous?
A: Solve 9x − 7 ≥ 0, so x ≥ 7/9. The interval of continuity is [7/9, ∞).
Q: Where is f(x) = ∛(8x − 3) continuous?
A: (−∞, ∞). Cube root functions are continuous for all real numbers.
Q: Evaluate lim(x→π) sin(3x − sin(6x)).
A: The inner expression 3x − sin(6x) is continuous everywhere. By direct substitution: sin(3π − sin(6π)) = sin(3π − 0) = sin(3π) = 0.
Continuity connects directly to the definition of the derivative. A function must be continuous at a point to be differentiable there (though the converse is false: continuous does not guarantee differentiable). The Intermediate Value Theorem, which you will use to prove that equations have solutions, requires continuity on a closed interval. Later, when you reach integration, the Fundamental Theorem of Calculus assumes continuity as well.
continuity, continuous function, discontinuity, removable discontinuity, hole in graph, jump discontinuity, infinite discontinuity, vertical asymptote, interval notation, domain of a function, polynomial continuity, rational function domain, square root domain, cube root domain, direct substitution, limit by continuity, MTH 132, Calculus I, Michigan State University, Section 1.8