What does it mean for a function to be continuous?

A continuous function has no breaks in its graph—you can trace it without lifting your pen. Intuitively, small changes in input produce small changes in output, with no sudden jumps, missing points, or explosions to infinity.

What is the formal definition of continuity at a point?

A function f is continuous at x = a if three conditions hold: f(a) is defined, the two-sided limit exists, and the limit equals the function value. The equation lim(x→a) f(x) = f(a) encapsulates all three requirements.

How is continuity defined on a closed interval?

On [a,b], the function must be continuous on the open interval (a,b), right-continuous at a (limit from right equals f(a)), and left-continuous at b (limit from left equals f(b)). Endpoints use one-sided continuity.

What are the types of discontinuities?

The four types are: removable (limit exists but doesn't equal f(a)), jump (one-sided limits exist but differ), infinite (at least one one-sided limit is ±∞), and oscillating (limit fails to exist due to oscillation).

What is a removable discontinuity?

A removable discontinuity occurs when the limit exists but the function is undefined or has a different value at that point. It appears as a hole in the graph and can be 'fixed' by redefining the function value to equal the limit.

What is a jump discontinuity?

A jump discontinuity occurs when both one-sided limits exist but are unequal. The function 'jumps' from one value to another. The floor function has jump discontinuities at every integer, jumping by 1 each time.

What is an infinite discontinuity?

An infinite discontinuity occurs when at least one one-sided limit equals ±∞. The function has a vertical asymptote at that point. For f(x) = 1/x at x = 0, both one-sided limits are infinite with opposite signs.

Which standard functions are continuous?

Polynomials are continuous everywhere. Rational functions are continuous except where the denominator is zero. Trig functions, exponentials, logarithms, and roots are continuous on their domains. For these, limits can be evaluated by direct substitution.

How do you use IVT to find roots?

If f is continuous on [a,b] with f(a) 0 (or vice versa), IVT guarantees f(c) = 0 for some c between a and b. A sign change means a root exists. The bisection method refines this by repeatedly halving the interval.

Continuity of Limits

Q: Are combinations of continuous functions continuous?

Yes. If f and g are continuous at a, then f+g, f−g, f·g, and f/g (when g(a)≠0) are continuous at a. Compositions are also continuous: if g is continuous at a and f is continuous at g(a), then f∘g is continuous at a.

The Intuitive Idea

The Formal Definition

Continuity on Intervals

Types of Discontinuities

Removable Discontinuity

Jump Discontinuity

Infinite Discontinuity

Oscillating Discontinuity

Continuous Functions — The Standard Library

Combinations of Continuous Functions

The Intermediate Value Theorem (IVT)

Using IVT to Locate Roots

Summary: Where Continuity Is the Hypothesis

Where Limits Meet Function Values

A function is continuous at a point when its limit there equals its actual value. No gap, no jump, no missing point—the function does exactly what its nearby behavior predicts.

This simple requirement has profound consequences. Continuous functions behave predictably: you can evaluate limits by direct substitution, the graph has no breaks, and values change gradually rather than abruptly.

Discontinuities occur when something fails. The function might be undefined at the point, the limit might not exist, or the limit might exist but differ from the function value. Classifying what goes wrong reveals the nature of the break and whether it can be repaired.

Key Terms

Continuity—

\lim_{x \to a} f(x) = f(a)

, three conditions in one equation

Discontinuity— removable, jump, infinite, or oscillating

Limit— the two-sided limit must exist for continuity to hold

One-Sided Limit— one-sided continuity at endpoints of closed intervals

Asymptote— infinite discontinuities produce vertical asymptotes

See All Calculus Definitions →

The Formal Definition

A function

f

is continuous at

x = a

if three conditions hold:

1.

f(a)

is defined — the function has a value at

a

\lim_{x \to a} f(x)

exists — the two-sided limit exists

3.

\lim_{x \to a} f(x) = f(a)

— the limit equals the function value

All three conditions must hold. The single equation

\lim_{x \to a} f(x) = f(a)

encapsulates all three: the right side requires

f(a)

to exist, the left side requires the limit to exist, and the equality requires them to match.

Continuity on Intervals

A function is continuous on an open interval

(a, b)

if it is continuous at every point in that interval.

For closed intervals

[a, b]

, the definition adjusts at endpoints where only one direction of approach is available:

$(a, b)$
$a$ : $\lim_{x \to a^+} f(x) = f(a)$
$b$ : $\lim_{x \to b^-} f(x) = f(b)$

One-sided continuity handles the boundaries where the domain restricts approach to a single direction.

Types of Discontinuities

Discontinuities are classified by which condition fails and how.

Removable discontinuity: The limit exists, but

f(a)

is undefined or

f(a) \neq \lim_{x \to a} f(x)

Jump discontinuity: Both one-sided limits exist but differ

Infinite discontinuity: At least one one-sided limit is

\pm\infty

Oscillating discontinuity: The limit fails to exist due to oscillation

Each type reveals different information about the function's behavior and determines whether the discontinuity can be "fixed."

Removable Discontinuity

A removable discontinuity occurs when the limit exists but differs from the function value (or the function is undefined).

Consider:

f(x) = \frac{x^2 - 1}{x - 1}

For

x \neq 1

, this simplifies to

x + 1

. The limit as

x \to 1

2

. But

f(1)

is undefined—division by zero.

The graph shows a hole at

(1, 2)

. The discontinuity is removable because redefining

f(1) = 2

fills the hole and makes the function continuous.

Removable discontinuities are "accidents" that can be repaired with a single point adjustment.

Jump Discontinuity

A jump discontinuity occurs when both one-sided limits exist but are unequal:

\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)

The function "jumps" from one value to another at

x = a

.

The floor function

\lfloor x \rfloor

jumps at every integer. At

x = 2

\lim_{x \to 2^-} \lfloor x \rfloor = 1 \qquad \lim_{x \to 2^+} \lfloor x \rfloor = 2

No single value of

f(2)

can equal both one-sided limits. The discontinuity cannot be removed—the gap is intrinsic to the function's definition.

Infinite Discontinuity

An infinite discontinuity occurs when the function blows up—at least one one-sided limit equals

\pm\infty

.

For

f(x) = \dfrac{1}{x}

x = 0

\lim_{x \to 0^-} \frac{1}{x} = -\infty \qquad \lim_{x \to 0^+} \frac{1}{x} = +\infty

The graph has a vertical asymptote at

x = 0

. No finite value can serve as

f(0)

because the function escapes to infinity from both sides.

Infinite discontinuities mark points where the function is fundamentally unbounded.

Oscillating Discontinuity

An oscillating discontinuity occurs when the function oscillates without settling, preventing the limit from existing.

The classic example is

f(x) = \sin(1/x)

x = 0

. As

x \to 0

, the argument

1/x \to \pm\infty

, causing sine to oscillate between

-1

and

1

infinitely often.

No single value

L

works because the function repeatedly swings away from any candidate. The limit does not exist—not because of a jump or blowup, but because of uncontrolled oscillation.

Discontinuity type	Which condition fails	Graph signal	Removable?
Removable	limit exists, but f(a) is undefined OR f(a) ≠ lim_{x → a} f(x)	a single hole in the curve at (a, L)	✓
Jump	both one-sided limits exist but differ — two-sided limit fails to exist	a step / gap between two finite levels	✗
Infinite	at least one one-sided limit is ±∞ — limit fails to exist (in the finite sense)	vertical asymptote at x = a	✗
Oscillating	function oscillates without settling — limit fails to exist	infinitely many swings as x → a (e.g. sin(1/x))	✗

Continuous Functions — The Standard Library

Many familiar functions are continuous on their entire domains:

Polynomials: continuous everywhere

Rational functions: continuous except where the denominator is zero

Trigonometric functions: $\sin x$ , $\cos x$ continuous everywhere; $\tan x$ , $\sec x$ , etc. continuous where defined

Exponential functions: $e^x$ , $a^x$ continuous everywhere

Logarithmic functions: $\ln x$ , $\log_a x$ continuous on $(0, \infty)$

Root functions: $\sqrt[n]{x}$ continuous on their domains

For these functions, evaluating limits reduces to direct substitution.

Function class	Continuous on	Possible discontinuities
Polynomial	all of ℝ	none
Rational p(x) / q(x)	{x : q(x) ≠ 0}	removable (if p and q share a root) or infinite (if only q vanishes)
Exponential a^x (a > 0)	all of ℝ	none
Logarithm ln x (and log_b x)	(0, ∞)	infinite at x = 0 (vertical asymptote; undefined for x ≤ 0)
Trig sin x, cos x	all of ℝ	none
Trig tan x, sec x, csc x, cot x	domain of the function (excludes its asymptotes)	infinite at each asymptote (periodic)
Root ⁿ√x	odd n: all of ℝ. even n: [0, ∞)	none in the interior; for even n the function is undefined for x < 0

Combinations of Continuous Functions

Continuity is preserved under standard operations. If

f

and

g

are continuous at

a

, then so are:

f + g \qquad f - g \qquad f \cdot g \qquad \frac{f}{g} \;(\text{if } g(a) \neq 0)

Compositions also preserve continuity: if

g

is continuous at

a

and

f

is continuous at

g(a)

, then

f \circ g

is continuous at

a

.

These limit rules carry over directly to continuity, since continuity is defined in terms of limits.

The Intermediate Value Theorem (IVT)

f

is continuous on

[a, b]

and

k

is any value between

f(a)

and

f(b)

, then

f(c) = k

for some

c

(a, b)

f(a) < k < f(b) \quad \Longrightarrow \quad f(c) = k \text{ for some } c \in (a, b)

Continuous functions hit every intermediate value—no skipping. The graph cannot jump from

f(a)

f(b)

without passing through every height in between.

This theorem guarantees existence. It says a

c

exists but does not tell you where or whether it is unique.

Using IVT to Locate Roots

A sign change guarantees a root. If

f

is continuous on

[a, b]

with:

f(a) < 0 \quad \text{and} \quad f(b) > 0

then

f(c) = 0

for some

c \in (a, b)

.

Example: Show that

x^3 - x - 1 = 0

has a solution in

(1, 2)

.

Let

f(x) = x^3 - x - 1

f(1) = 1 - 1 - 1 = -1 < 0

f(2) = 8 - 2 - 1 = 5 > 0

Since

f

is continuous and changes sign, IVT guarantees a root between

1

and

2

.

The bisection method refines this: repeatedly halve the interval, keeping the half where the sign change occurs, to approximate the root to any desired precision.

Summary: Where Continuity Is the Hypothesis

Continuity isn't just a property a function might happen to have; it's a hypothesis that unlocks a long list of useful results across the limits subtree and beyond. The table below collects five places where this page's definition of continuity does the work — from the everyday move of direct substitution for limits, through the algebraic and composition rules, to the Intermediate Value Theorem. In each row, the required form of continuity is the condition that turns the right-hand result into a guarantee.

Where continuity appears	Required form	What it enables
Evaluating limits by substitution	f continuous at a	lim_{x → a} f(x) = f(a) — the limit reduces to plugging in
Algebraic limit rules	f and g continuous at a	f ± g, f · g, and f / g (where g(a) ≠ 0) inherit continuity at a
Composition rule for limits	g continuous at a; f continuous at g(a)	f∘g continuous at a; limits pass through the outer function
Analysis on a closed interval	continuous on [a, b] via one-sided continuity at endpoints	the function is well-behaved across the full closed interval — endpoints included
Intermediate Value Theorem	continuous on [a, b]	f attains every value between f(a) and f(b) — guarantees roots when the sign changes