Continuity of Limits

The Intuitive Idea

The Formal Definition

Continuity Notation

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

Continuity FAQ

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

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The Intuitive Idea

A continuous function has no breaks in its graph. You can trace the curve without lifting your pen. The function value at each point matches what the surrounding values predict.

Intuitively, small changes in input produce small changes in output. There are no surprises—no sudden jumps, no missing points, no explosions to infinity.

This matches everyday experience. Temperature varies continuously throughout the day. Position changes continuously as you walk. Discontinuities—sudden jumps or gaps—signal something unusual: a switch being flipped, a boundary being crossed.

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

Continuity at a Point

f \text{ continuous at } a \iff \lim_{x \to a} f(x) = f(a)

Learn more about this formula: Continuity at a Point →

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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.

Condition 3 fails: lim f(x) ≠ f(1)

The checker tests the three conditions in order: here f(1) is defined and the two-sided limit exists, but the marked point sits away from the limit, so the equality in condition 3 fails and the verdict is discontinuous. One relocated point would make the function continuous. Run the three-step check on any example on the continuity checker.

Each way the three conditions can fail has a name, and the next sections take them in turn.

Continuity Notation

Notation

Continuity Notation

What the single equation packs in, its one-sided halves, and the class mark that turns “continuous on” into set membership.

Base

\lim

and DNE — two-sided limits; the superscripts — one-sided limits;

\pm\infty

— limits at infinity.

\lim\limits_{x \to a} f(x) = f(a)

The limit of f at a equals f of a

Three claims in one equation —

f(a)

exists, the two-sided limit exists, they match. Fail any one and the equation is false or meaningless.

CasesInterior point — as written. Open interval

(a,b)

— the equation at every point. Closed

[a,b]

— endpoints switch to the one-sided form below.

Also writtenEpsilon-delta:

|x - a| < \delta \Rightarrow |f(x) - f(a)| < \varepsilon

. Note the missing

0 <

— unlike a limit's punctured

0 < |x - a|

, continuity lets

x = a

in.

Do not confuseA computation rule. “Evaluate by substitution” is what continuity licenses, not what defines it — quoting substitution to prove continuity runs in a circle.

\lim\limits_{x \to a^{+}} f(x) = f(a)

f is right-continuous at a

The continuity equation restricted to one side. The superscript machinery belongs to one-sided limits; this page adds the equality with

f(a)

CasesLeft endpoint of

[a,b]

— right-continuity is all that is asked; the right endpoint mirrors it with

b^{-}

. Every cumulative distribution function is right-continuous by convention — this equation is its defining property.

Also written

f(a^{+}) = f(a)

, in the one-sided value shorthand.

Do not confuseFull continuity. At an interior point one-sided continuity is strictly weaker — the floor function is right-continuous everywhere yet jumps at every integer, as in Jump Discontinuity below.

f \in C[a,b]

f is in C of a, b

Class membership:

C[a,b]

is the set of all functions continuous on

[a,b]

C(I)

on an interval

I

. Theorem hypotheses shrink to

f \in C[a,b]

instead of a sentence.

Cases

C^0

— the same class with its ladder rung showing; the ladder

C^1, C^2, \ldots, C^\infty

counts continuous derivatives and is climbed on higher-order derivatives.

Also written

C^0[a,b]

;

C(X)

in topology, for continuous functions on a space

X

Do not confuseFunction arguments. The brackets are the interval's own —

C[a,b]

is not

C

evaluated at

a

and

b

Same glyph elsewhereRoman

C

also counts combinations

C(n,r)

, and blackboard

\mathbb{C}

names the complex numbers — unrelated jobs for one letter.

All calculus symbols →↑ Limits

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:

Continuous on $(a, b)$
Right-continuous at $a$ : $\lim_{x \to a^+} f(x) = f(a)$
Left-continuous at $b$ : $\lim_{x \to b^-} f(x) = f(b)$

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."

The four types sort into two groups by one question: does the limit at the point exist? If it does, the fault is confined to a single value and can be repaired by redefining

f(a)

. If it does not, no choice of

f(a)

can help, and the three ways the limit can fail give the three remaining types.

DownloadPNG2×PDF

Removablethe limit exists; the fault is a single value, so redefining $f(a)$ makes the function continuous
- Hole $f(a)$ is undefined $(x^2 - 1)/(x - 1)$ at $x = 1$ : limit $2$ , no value
- Misplaced value $f(a)$ exists but differs from the limitthe point sits off the curve
Non-removablethe limit does not exist; no single value of $f(a)$ can fix it
- Jumpboth one-sided limits exist but differ $\lfloor x \rfloor$ at $x = 2$ : $1$ from the left, $2$ from the right
- Infiniteat least one one-sided limit is $\pm\infty$ $1/x$ at $x = 0$
- Oscillatingthe limit fails to exist because the function oscillates without settling $\sin(1/x)$ at $x = 0$

the limit exists: one value repairs itthe limit does not exist: nothing repairs it

Four discontinuities, two groups: repairable or not·/calculus/limits/continuityLearn Math Class

The group is settled before

f(a)

is even looked at: compute the two one-sided limits. If they agree and are finite, the discontinuity is removable and only

f(a)

needs checking. If they disagree, it is a jump; if one of them is infinite, it is infinite; if neither settles on a value, it is oscillating. Each type is taken up in its own section below.

Removable Discontinuity

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

Consider:

Removable Discontinuity

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.

Hole at x = 1: removable

The limit at 1 exists and equals 2, but the function has no value there; filling the single missing point would make the curve continuous, which is why the discontinuity is called removable. The simplified formula x + 1 is exactly that repaired function. Compare the hole with the repaired curve on the continuity checker.

Removable discontinuities are the mild case: the function is continuous except for a bookkeeping error at one point.

Jump Discontinuity

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

Jump Discontinuity

\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

Jump Discontinuity

\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.

Jump at x = 0: left limit 0, right limit 1

From the left the function approaches 0, from the right it approaches 1, and no single value could bridge the gap: the two-sided limit does not exist, so condition 2 fails. The size of the jump is the difference of the two one-sided limits. Read the two one-sided values on the continuity checker.

Jumps are the natural discontinuities of piecewise definitions and of the floor and ceiling functions.

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

Infinite Discontinuity

\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.

Asymptote at x = 0: the limit is infinite

Approaching 0 the function runs off the top or bottom of the picture: at least one one-sided limit is infinite, so there is no finite limit and no value that could be assigned. The vertical line x = 0 is the asymptote the curve hugs. Watch the readings blow up on the continuity checker.

Infinite discontinuities mark the vertical asymptotes of rational functions.

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))	✗

sin(1/x) near 0: no limit at all

The probes at ±0.5 read sin 2 ≈ 0.91, but moving them inward the readings swing between −1 and 1 faster and faster, never settling: the limit fails to exist not by running away but by refusing to choose. No value of f(0) could make this continuous. Squeeze the probes and watch the readings swing on the limit explorer.

Oscillating discontinuities are rare in practice but show that the limit can fail without any value being infinite.

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:

Continuity Preserved Under Operations

f, g \text{ continuous at } a \implies f \pm g, \; cf, \; f \cdot g, \; f/g \;(g(a) \neq 0), \; f \circ g \text{ continuous at } a

Learn more about this formula: Continuity Preserved Under Operations →

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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)

Intermediate Value Theorem

f \text{ continuous on } [a,b], \; k \text{ between } f(a) \text{ and } f(b) \implies \exists\, c \in (a,b) \text{ with } f(c) = k

Learn more about this formula: Intermediate Value Theorem →

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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.

Every height in between is reached

f(x) = x³ − 3x is continuous on [−2, 2.2], starts at f(−2) = −2 and ends at about 4.05. Any level between those two heights, such as the dashed k = 1, must be crossed: the unbroken curve cannot get from below the line to above it without meeting it. Here it meets it three times. The theorem only promises at least one such c, not where it is or how many there are.

Without continuity the guarantee fails: a jump can skip a level entirely.

Using IVT to Locate Roots

A sign change guarantees a root. If

f

is continuous on

[a, b]

with:

Using IVT to Locate Roots

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.

Trapping a root by halving

The section’s f(x) = x³ − x − 1 is negative at 1 and positive at 2, so a root lies between. f(1.5) ≈ 0.875 is positive, so the root is in [1, 1.5]; f(1.25) ≈ −0.30 is negative, so it is in [1.25, 1.5], between the dashed lines. Each step halves the interval that must contain the root; the root itself is about 1.3247.

Ten halvings shrink [1, 2] to an interval shorter than 0.001.

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

Continuity FAQ

What does f ∈ C[a,b] mean?

It says f belongs to Ca,b], the set of all functions continuous on the closed interval [a, b]. The brackets are the interval’s own, not arguments to C. Writing it this way shrinks a theorem hypothesis from a sentence to one symbol. The same class is also written C⁰ — the bottom rung of the ladder C¹, C², …, C^∞, which counts [continuous derivatives.Read more →

How does the epsilon-delta definition of continuity differ from the one for limits?

By a single missing condition. A limit requires 0 < |x − a| < δ, deliberately puncturing the point so that x never equals a. Continuity drops the 0 <, writing |x − a| < δ implies |f(x) − f(a)| < ε, which lets x = a into the argument. That inclusion is exactly what forces the value at a to agree with the surrounding behavior.Read more →

Can you prove a function is continuous by substitution?

No — that reasoning runs in a circle. Evaluating a limit by substituting the point is something continuity licenses, not something that establishes it; substitution works precisely because the function is already known to be continuous there. To prove continuity, check the three conditions directly or cite a theorem, such as the continuity of polynomials or of sums and products of continuous functions.Read more →