Continuity on Intervals

When we study continuity in calculus, we often need to go beyond asking whether a function is continuous at a single point and instead ask: is the function continuous across an entire interval? A function is said to be continuous on an interval if it is continuous at every point within that interval. This idea is fundamental to many powerful theorems in calculus, including the Intermediate Value Theorem and the Extreme Value Theorem, both of which require continuity on a closed interval. There are two main types of intervals to consider: open intervals (a, b), which exclude the endpoints, and closed intervals [a, b], which include both endpoints. The rules for continuity differ slightly at interior points versus at endpoints. At an interior point c in (a, b), we require the standard three-part definition: the function value f(c) exists, the two-sided limit lim_{x→c} f(x) exists, and these two values are equal. At a left endpoint a of a closed interval, we only require the right-hand limit to equal the function value, and at a right endpoint b, we only require the left-hand limit to equal the function value. Understanding continuity on intervals allows mathematicians and scientists to model real-world phenomena where quantities change smoothly over a range of values — think of the position of a moving object, the temperature along a metal rod, or the voltage across a circuit. Once we confirm a function is continuous on an interval, we unlock powerful analytical tools that would not be valid otherwise. This article explores the precise definitions, key tests, and worked examples for determining continuity on intervals.

Key takeaways

Recall Point Continuity
Continuity on an Open Interval (a, b)
Continuity on a Closed Interval [a, b]
Using Algebra to Identify Discontinuities
Applying the Intermediate Value Theorem (IVT)

Recall Point Continuity

A function f is continuous at an interior point c if: (1) f(c) is defined, (2) lim_{x→c} f(x) exists (the two-sided limit exists), and (3) lim_{x→c} f(x) = f(c). All three conditions must hold simultaneously. A failure at any one condition creates a discontinuity — a removable discontinuity if the limit exists but doesn't equal f(c), a jump discontinuity if the one-sided limits exist but differ, or an infinite discontinuity if the limit is ±∞.

Continuity on an Open Interval (a, b)

A function f is continuous on the open interval (a, b) if and only if it is continuous at every point c with a < c < b. The endpoints a and b are not included, so we impose no condition there. For example, f(x) = 1/x is continuous on the open interval (0, ∞) because at every c > 0, f(c) = 1/c is defined, the limit equals 1/c, and there is no disruption. However, 1/x is NOT continuous on (−1, 1) because it is undefined at x = 0, which lies inside the interval.

Continuity on a Closed Interval [a, b]

A function f is continuous on the closed interval [a, b] if: (1) it is continuous at every interior point c in (a, b), (2) it is right-continuous at the left endpoint a, meaning lim_{x→a⁺} f(x) = f(a), and (3) it is left-continuous at the right endpoint b, meaning lim_{x→b⁻} f(x) = f(b). This one-sided condition at endpoints reflects that we cannot approach a or b from outside [a, b]. For example, f(x) = √x is continuous on [0, ∞) because at x = 0 we check only the right-hand limit: lim_{x→0⁺} √x = 0 = f(0). ✓

Using Algebra to Identify Discontinuities

Most standard functions — polynomials, sine, cosine, exponentials — are continuous everywhere on their domains. Rational functions f(x) = p(x)/q(x) are continuous everywhere except where q(x) = 0. To check continuity on an interval, first identify where the denominator is zero or where square roots, logarithms, or other restricted operations create undefined values. If none of these 'trouble spots' fall inside the interval, the function is continuous there. For piecewise functions, also check the boundaries between pieces by computing one-sided limits from both sides.

Applying the Intermediate Value Theorem (IVT)

One immediate payoff of continuity on a closed interval is the IVT: if f is continuous on [a, b] and N is any number strictly between f(a) and f(b), then there exists at least one c in (a, b) such that f(c) = N. This guarantees that a continuous function must hit every value between its endpoint values — it cannot 'jump over' any number. For example, since f(x) = x³ − x − 2 is a polynomial (continuous everywhere), and f(1) = −2 < 0 while f(2) = 4 > 0, the IVT guarantees a root somewhere in (1, 2).

The Key Equation

f \text{ is continuous on } [a,b] \iff \begin{cases} \lim_{x \to c} f(x) = f(c) & \forall\, c \in (a,b) \\ \lim_{x \to a^+} f(x) = f(a) & \text{(right-continuous at } a) \\ \lim_{x \to b^-} f(x) = f(b) & \text{(left-continuous at } b) \end{cases}

f — Function (dimensionless): The real-valued function whose continuity is being tested on the interval.
[a, b] — Closed Interval (dimensionless): The set of all real numbers x satisfying a ≤ x ≤ b, including both endpoints a and b.
\lim_{x \to c} — Two-sided limit (dimensionless): The limit of f(x) as x approaches c from both the left and the right simultaneously.
\lim_{x \to a^+} — Right-hand limit (dimensionless): The limit of f(x) as x approaches a from values strictly greater than a.
\lim_{x \to b^-} — Left-hand limit (dimensionless): The limit of f(x) as x approaches b from values strictly less than b.

Worked Example

Problem. Determine whether the piecewise function f(x) = { x² + 1, if x ≤ 2 ; 3x − 1, if x > 2 } is continuous on the closed interval [0, 5].

\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = f(2) = 5 \implies f \text{ is continuous on } [0,5]

Answer. The function f is continuous on the closed interval [0, 5] because both polynomial pieces are individually continuous, and at the joining point x = 2, the left-hand limit, right-hand limit, and function value all equal 5.

Figure

Diagram

Origins · 1 minAugustin-Louis Cauchy and how this was discovered

Augustin-Louis Cauchy

1789–1857 · French

Cauchy is credited with giving the first rigorous definition of continuity and limits in his landmark textbook Cours d'analyse (1821). Before Cauchy, mathematicians used intuitive notions of continuous functions without precise definitions. He formalized the concept of a function being continuous at a point and on an interval, laying the groundwork for all of modern analysis. His epsilon-delta framework, later refined by Weierstrass, is the foundation of how we define limits and continuity today. Cauchy was extraordinarily prolific, publishing nearly 800 papers and books during his lifetime — so many that the French Academy of Sciences eventually had to limit the number of papers any one mathematician could submit per week, a rule informally called the 'Cauchy rule.'

Continuity at a Point — Continuity on an interval is built directly from the definition of continuity at a single point; you must master the three-part point definition before extending it to intervals.
Intermediate Value Theorem — The IVT is one of the most important consequences of continuity on a closed interval — it guarantees the existence of function values between any two outputs, which is used to locate roots.
Types of Discontinuity — Identifying removable, jump, and infinite discontinuities helps pinpoint exactly where and why a function fails to be continuous on a given interval.
One-Sided Limits — Continuity at the endpoints of a closed interval requires one-sided limits; understanding left- and right-hand limits is essential for endpoint analysis.
Extreme Value Theorem — Like the IVT, the Extreme Value Theorem assumes continuity on a closed interval to guarantee that a function attains its absolute maximum and minimum values.