Continuity of a Function: Definition & Examples

What Is Continuity Of A Function?

Continuity of a function is the precise statement that a function has no break at a point: as the input creeps toward $a$, the output creeps toward the actual value $f(a)$, with nothing missing in between. A function is continuous at $x = a$ exactly when all three of the following hold.

$$\begin{aligned} &\text{(1)} \quad f(a) \ \text{is defined} \ &\text{(2)} \quad \lim_{x \to a} f(x) \ \text{exists} \quad (\text{left limit} = \text{right limit}) \ &\text{(3)} \quad \lim_{x \to a} f(x) = f(a) \end{aligned}$$

Each condition rules out one way a graph can break. Condition (1) rules out a missing point (a hole where the function has no value). Condition (2) rules out a jump, where the graph arrives at two different heights from the two sides. Condition (3) rules out a stray point, where the limit and the plotted value disagree.

Only when the limit of a function both exists and matches $f(a)$ does the curve pass through its own limiting height with no interruption.

A function that satisfies all three at every point of an interval is called continuous on that interval. The opposite, a point where any condition fails, is a point of discontinuity, and the different failures are sorted into named types of discontinuity.

What Are The Three Conditions For Continuity Of A Function?

The three conditions are best read as a checklist you run in order, because each one depends on the one before it.

• Condition 1, the value exists. You cannot ask whether a graph passes through a point if the point is not there. If $f(a)$ is undefined, the function is discontinuous at $a$ before you even look at the limit.

• Condition 2, the limit exists. The one-sided limits $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ must both exist and be equal. If they disagree, the two-sided limit does not exist and the graph jumps.

• Condition 3, the two agree. Even with a value and a limit in hand, continuity needs them to be the same number. If $\lim_{x \to a} f(x) \neq f(a)$, the plotted point sits off the curve and the graph has a stray hole.

Geometrically the whole idea is one sentence: the height the graph is heading toward is the height it actually reaches. The algebra of the three conditions is just that sentence written carefully.

How Do You Check Continuity Of A Function At A Point?

Run the three-condition checklist. The three worked examples below are the three cases you meet most: a hole you can patch, a seam that holds, and a step that breaks.

Example 1: A removable hole. Is $f(x) = \dfrac{x^2 - 1}{x - 1}$ continuous at $x = 1$?

Start with condition 1. Substituting $x = 1$ gives $\dfrac{0}{0}$, which is undefined, so $f(1)$ does not exist and condition 1 already fails. The function is discontinuous at $x = 1$.

The limit, however, is perfectly healthy. Factor and cancel for $x \neq 1$:

$$\frac{x^2 - 1}{x - 1} = \frac{(x - 1)(x + 1)}{x - 1} = x + 1, \qquad \lim_{x \to 1} f(x) = 1 + 1 = 2$$

Because the limit exists ($2$) but the value is missing, this is a removable discontinuity: defining $f(1) = 2$ patches the hole and makes the function continuous. See removable discontinuity for the full patching rule.

Final answer: Not continuous at $x = 1$; removable, and $f(1) := 2$ repairs it.

Example 2: A piecewise seam. Is $f$ continuous at $x = 2$, where

$$f(x) = \begin{cases} x^2 + 1, & x \le 2 \ 4x - 3, & x > 2 \end{cases}$$

Check the value: $f(2) = 2^2 + 1 = 5$, so condition 1 holds. Now the one-sided limits at the seam:

$$\lim_{x \to 2^-} f(x) = 2^2 + 1 = 5, \qquad \lim_{x \to 2^+} f(x) = 4(2) - 3 = 5$$

The two sides agree, so $\lim_{x \to 2} f(x) = 5$ and condition 2 holds. Finally $\lim_{x \to 2} f(x) = 5 = f(2)$, so condition 3 holds. All three pass.

Final answer: Continuous at $x = 2$.

Example 3: A step that breaks. Is $f$ continuous at $x = 0$, where

$$f(x) = \begin{cases} 1, & x < 0 \ 2, & x \ge 0 \end{cases}$$

The value exists: $f(0) = 2$, so condition 1 holds and it is tempting to stop there. Check the limits before you do:

$$\lim_{x \to 0^-} f(x) = 1, \qquad \lim_{x \to 0^+} f(x) = 2$$

The left and right limits disagree, so $\lim_{x \to 0} f(x)$ does not exist and condition 2 fails. This is a jump discontinuity, discussed further under jump discontinuity.

Final answer: Not continuous at $x = 0$; the two-sided limit does not exist.

What Does Continuity Of A Function On An Interval Mean?

A function is continuous on an open interval $(a, b)$ when it is continuous at every point inside it. Closed intervals need one extra idea, because the endpoints only have one side.

At an endpoint you use a one-sided limit. A function is continuous from the right at $a$ when $\lim_{x \to a^+} f(x) = f(a)$, and continuous from the left at $b$ when $\lim_{x \to b^-} f(x) = f(b)$. Putting these together:

$$f \ \text{is continuous on } [a, b] \iff \begin{cases} f \ \text{is continuous on } (a, b), \ \lim_{x \to a^+} f(x) = f(a), \ \lim_{x \to b^-} f(x) = f(b). \end{cases}$$

The reason is simply that no points exist to the left of $a$ or to the right of $b$, so demanding a two-sided limit there would be impossible. The function $f(x) = \sqrt{x}$ is the standard example: it is continuous on $[0, \infty)$, right-continuous at the endpoint $0$, even though nothing lies to the left of $0$ to test.

What Is The Algebra Of Continuous Functions?

Continuity survives the ordinary operations of arithmetic, which is why you almost never check continuity from the definition in practice. If $f$ and $g$ are both continuous at $x = a$, then so are the combinations below.

• Sum and difference: $f + g$ and $f - g$ are continuous at $a$.

• Product: $fg$ is continuous at $a$, and so is any constant multiple $k f$.

• Quotient: $\dfrac{f}{g}$ is continuous at $a$ provided $g(a) \neq 0$. The single exclusion is where the denominator vanishes.

• Composite: if $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then $f\big(g(x)\big)$ is continuous at $a$.

These rules let you build continuity of a function from small pieces. Because $x$ and every constant are continuous everywhere, the sum-product rule immediately makes every polynomial continuous everywhere, and the quotient rule makes every rational function continuous except where its denominator is zero.

Table: Which standard functions are continuous, and where.

The last three matter for exam marks: a function can only be continuous where it is defined, so $\tan x$, $\ln x$, and $\sqrt{x}$ are continuous on their domains, never at points their domains exclude.

How Does Continuity Of A Function Lead To The Intermediate Value Theorem?

The reward for a function being continuous on a closed interval is the Intermediate Value Theorem (IVT). It says that a continuous function cannot skip any output value between its endpoints.

Formally: if $f$ is continuous on $[a, b]$ and $N$ is any number between $f(a)$ and $f(b)$, then there is at least one $c$ in $[a, b]$ with $f(c) = N$. The theorem needs continuity, exactly because a jump would let the graph leap over $N$ without ever equalling it.

Worked payoff: show $f(x) = x^3 + x - 1$ has a root in $[0, 1]$.

The function is a polynomial, so it is continuous on $[0, 1]$. Evaluate the endpoints:

$$f(0) = -1 < 0, \qquad f(1) = 1^3 + 1 - 1 = 1 > 0$$

Since $0$ lies between $f(0) = -1$ and $f(1) = 1$, the IVT guarantees some $c$ in $(0, 1)$ with $f(c) = 0$. Continuity is doing all the work here: without it, a function could pass from negative to positive without ever hitting zero. The full statement lives at intermediate value theorem.

Why Does Continuity Of A Function Matter?

Continuity is the quiet assumption behind most of the theorems students meet later, so it earns its place as a definition rather than a footnote.

• It is what lets you substitute. For a continuous function, $\lim_{x \to a} f(x) = f(a)$, so "take the limit" and "plug in the number" give the same answer. Every quick limit you evaluate by substitution is secretly using continuity.

• It is required before differentiation. A function must be continuous at a point to have a derivative there, though continuity alone is not enough. That one-way relationship is the whole subject of continuity and differentiability.

• It powers existence theorems. The Intermediate Value Theorem, the Extreme Value Theorem, and the Mean Value Theorem all open with the words "let $f$ be continuous." Drop continuity and each one has an easy counterexample.

Seen this way, continuity is less about drawing tidy graphs and more about a promise: the function holds no surprises between the points you can see, so the tools of calculus are safe to use on it.

Who Discovered The Idea Of Continuity Of A Function?

Continuity felt obvious for centuries ("draw it without lifting the pen") long before anyone pinned it down with limits. The precise definition arrived in the early 1800s, and from an unexpected source.

Two names to keep with the topic:

• Augustin-Louis Cauchy (1789–1857, France) defined continuity and the limit with the care that made calculus rigorous, in his Cours d'analyse of 1821.

• Karl Weierstrass (1815–1897, Germany) sharpened the definition into the precise "epsilon-delta" language still taught today, closing the last gaps left by intuition.

Where Is Continuity Of A Function Used In The Real World?

Continuity is the mathematical version of "nothing teleports," so it shows up wherever a quantity changes without sudden gaps.

• Physics and motion: the position of a moving object is a continuous function of time. It cannot vanish from one spot and reappear in another, which is exactly the no-jump condition.

• Engineering and signals: a stable temperature, voltage, or pressure reading is modelled as continuous, and a genuine discontinuity flags a fault such as a broken sensor or a cut wire.

• Economics: many demand and cost curves are treated as continuous so that a small change in price produces a small change in quantity, which is what lets marginal analysis work at all.

• Control systems and robotics: a robot arm follows a continuous path so it moves smoothly between positions instead of snapping, and planners deliberately design continuous trajectories.

• Computer graphics: smooth shading and morphing rely on continuous functions of position and time, so surfaces blend without visible seams.

Across every one of these, the same idea does the work: outputs that stay close whenever inputs stay close, which is continuity of a function stated in plain language.

What Are The Most Common Mistakes With Continuity Of A Function?

These four errors account for most lost marks on continuity, and each matches a question real students ask on r/calculus, Quora, and AP review guides such as Albert.io's discontinuity walkthroughs.

Checking only that $f(a)$ is defined.

Where it slips in:

A student sees that the function has a value at $a$, declares it continuous, and moves on.

Don't do this:

Do not stop after condition 1. A value can exist while the limit jumps right past it, as in the step function where $f(0) = 2$ but the sides disagree.

The correct way:

Run all three conditions in order: value exists, limit exists, and the two are equal. Continuity needs every one.

Ignoring the one-sided limits.

Where it slips in:

A student computes a single limit at $a$ without separating the left and right sides, and misses that they disagree.

Don't do this:

Do not assume the two-sided limit exists. At a seam or a step, only the one-sided limits reveal a jump.

The correct way:

Compute $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ separately. The two-sided limit exists only when they are equal, which is what condition 2 demands.

Confusing "the limit exists" with "the function is continuous."

Where it slips in:

A student finds that $\lim_{x \to a} f(x)$ exists and concludes the function is continuous there, even when $f(a)$ is missing or different.

Don't do this:

Do not treat a healthy limit as the finish line. A removable discontinuity has a perfectly good limit and is still not continuous.

The correct way:

Compare the limit with the value. For $\dfrac{x^2 - 1}{x - 1}$ at $x = 1$ the limit is $2$ but $f(1)$ is undefined, so it is discontinuous until you define $f(1) = 2$.

Forgetting the domain of $\tan$, $\ln$, and rational functions.

Where it slips in:

A student calls $\tan x$ or $\ln x$ "continuous everywhere" and applies theorems across a point the function never reaches.

Don't do this:

Do not ignore where a function is undefined. A function cannot be continuous at a point outside its domain.

The correct way:

State the domain first. $\tan x$ is continuous except at $x = \tfrac{\pi}{2} + n\pi$, $\ln x$ only for $x > 0$, and a rational function only where its denominator is non-zero.

Practice Problems On Continuity Of A Function

Work each one, then check against the answer. Answers are verified.

1. Is $f(x) = \dfrac{x^2 - 9}{x - 3}$ continuous at $x = 3$?
   (Answer: No. $f(3)$ is undefined, but $\lim_{x \to 3} f(x) = 6$, so it is a removable discontinuity.)

2. Find $k$ so that $f(x) = \begin{cases} kx + 1, & x \le 1 \ 3x, & x > 1 \end{cases}$ is continuous at $x = 1$.
   (Answer: value and left limit give $k + 1$, right limit gives $3$, so $k + 1 = 3$ and $k = 2$.)

3. Is $f(x) = \lvert x \rvert$ continuous at $x = 0$?
   (Answer: Yes. $f(0) = 0$ and $\lim_{x \to 0^-} = \lim_{x \to 0^+} = 0$.)

4. Where is $f(x) = \dfrac{1}{x - 4}$ discontinuous?
   (Answer: only at $x = 4$, an infinite discontinuity; continuous everywhere else. See infinite discontinuity.)

5. Is the step $f(x) = \lfloor x \rfloor$ continuous at $x = 2$?
   (Answer: No. $\lim_{x \to 2^-} = 1$ and $\lim_{x \to 2^+} = 2$, a jump discontinuity.)

6. Show $f(x) = x^3 - 2x - 5$ has a root in $[2, 3]$.
   (Answer: it is continuous, $f(2) = -1 < 0$ and $f(3) = 16 > 0$, so by the IVT a root lies in $(2, 3)$.)

Where Should You Go Next After Continuity Of A Function?

Continuity is the gateway from limits into the rest of calculus, and several natural doors open from here.

1. Continuity and differentiability. See exactly how continuity is the entry requirement for a derivative, and why the reverse can fail.

2. Differentiability vs continuity. Line up the two ideas side by side, with the corner and cusp cases that separate them.

3. Types of discontinuity. Sort every break into removable, jump, or infinite, and learn which one you are looking at.

If your child is meeting continuity of a function for the first time, a live Bhanzu trainer teaches it from the three-condition checklist up, so limits, continuity, and derivatives feel like one connected story, in the Bhanzu math program.