Math · Calculus I · Concept
Continuity and the Intermediate Value Theorem
A continuous function has no holes, jumps or vertical asymptotes: f is continuous at x = a when f(a) is defined, the limit of f(x) as x approaches a exists, and the two are equal. On a closed interval, continuity gives the Intermediate Value Theorem (IVT): the function takes every value between f(a) and f(b), which proves that an equation has a solution.
Three conditions
f is continuous at a when f(a) is defined, the limit of f(x) as x approaches a exists, and the limit equals f(a). Each condition can fail on its own, so check them in order.
Kinds of discontinuity
When a condition fails, the way it fails names the discontinuity.
| Kind | What happens | Example |
|---|---|---|
| Removable | The limit exists, but f(a) is missing or different: a hole | (x² − 9)/(x − 3) at x = 3 |
| Jump | The one-sided limits exist but differ | |x|/x at x = 0 |
| Infinite | A one-sided limit is infinite: a vertical asymptote | 1/(x − 3)² at x = 3 |
Continuous functions
Polynomials are continuous everywhere. Rational functions, roots, exponentials, logarithms and the trig functions are continuous at every point of their domains. Sums, products, compositions and quotients with nonzero denominators of continuous functions are continuous.
Continuity on an interval
f is continuous on an open interval when it is continuous at every point of it. At the ends of a closed interval [a, b] only one side exists, so f needs the one-sided limits there: from the right at a and from the left at b.
Piecewise functions
Each piece is usually continuous on its own interval, so only the break points need checking. At a break point, the left piece and the right piece must approach the same height, and that height must be the function value.
The Intermediate Value Theorem
If f is continuous on [a, b] and N is any number between f(a) and f(b), then f(c) = N for at least one c in (a, b). A continuous graph cannot get from one height to another without passing through every height in between.
Trapping a root
If f(a) and f(b) have opposite signs, some c between them has f(c) = 0. Halving the interval and keeping the half where the sign changes, the bisection method, traps the root as tightly as you like: each halving gains about one binary digit.
What the theorem does not say
It promises at least one c, not exactly one, and gives no formula for it. Without continuity the conclusion can fail: 1/x is −1 at x = −1 and 1 at x = 1, but it is never 0.
Common mistakes
- Checking only that the limit exists: a removable hole has a limit but no matching function value.
- Checking one side at a break point: both one-sided limits must equal the function value.
- Applying the Intermediate Value Theorem across a break, such as 1/x on [−1, 1].
- Reading the theorem as giving exactly one solution: it guarantees at least one.
Key terms
- Continuity
- A function is continuous at a point when it is defined there and its limit there equals that value, so the graph has no hole, jump or break. On an interval this must hold at every point, using one-sided limits at closed ends.
- Discontinuity
- A point where a function isn’t continuous: a hole, a jump or a vertical asymptote. A plot with too few points can hide one or seem to show one that isn’t there.
- Removable discontinuity
- A hole in a graph: the function has a limit at the point but is undefined there or has a different value. Redefining the value to match the limit removes the hole.
- Jump discontinuity
- A point where the left-hand and right-hand limits both exist but are different, so the graph jumps. Changing the value at the point can’t fix it.
- Vertical asymptote
- A vertical line x = a that the graph approaches as the function’s values grow without bound near a, often where a denominator is zero. The graph is not joined across it.
- One-sided limit
- The limit as x approaches a from the left only (x → a⁻) or from the right only (x → a⁺). The two-sided limit exists exactly when both one-sided limits exist and are equal.
- Intermediate Value Theorem
- If f is continuous on [a, b] and N is any number between f(a) and f(b), then f(c) = N for at least one c in (a, b). It guarantees that a solution exists without saying where.
- Bisection
- Finding a root by repeatedly halving an interval where a continuous function changes sign, keeping the half where the sign change is. Each step halves the uncertainty.
Work through an example
Find the value of k that makes f continuous at x = 2, where f(x) = x² + k for x < 2 and f(x) = 3x − 1 for x ≥ 2.
Make a piecewise function continuous →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
See the pieces meet in Graph Check the left limit in Limits Solve for k in Math Open worked example on a board Intermediate Value Theorem in Math ReferenceYour existing work stays on this device. Examples open as editable copies.