What does it mean for a function to be continuous at a point?

A function f is continuous at point a when three conditions are satisfied: f(a) is defined, \lim_{x \to a} f(x) exists, and they are equal. Geometrically, this means you can draw the function through that point without lifting your pencil.

How do you check if a piecewise function is continuous at a breakpoint?

Use one-sided limits. Calculate \lim_{x \to a^{-}} f(x) using the left piece and \lim_{x \to a^{+}} f(x) using the right piece. If both one-sided limits are equal and match f(a), the function is continuous at the breakpoint.

Is every polynomial continuous?

Yes. All polynomial functions are continuous everywhere in their domain because they are sums and products of the simple function f(x) = x, which is continuous. This means you can always evaluate a polynomial's limit by direct substitution.

What is a removable discontinuity and how is it different from a jump discontinuity?

A removable discontinuity occurs when the limit exists but the function value is either undefined or different from the limit. A jump discontinuity occurs when the left and right limits exist but are not equal. Removable discontinuities can sometimes be 'fixed' by redefining the function value, whereas jump discontinuities cannot.

Continuity

Continuity tells you when a function has no jumps or breaks, so you can safely evaluate limits and apply theorems about function behavior.

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

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Type the problem. The solver will use Continuity where Continuity is the right tool, and tell you when it is not.

What each symbol means

What Continuity takes

Symbol	Meaning
$f$	The function being analyzed; it maps input values to output values, and if you confuse it with its value at a single point, you will miss the limit component.
$x$	The variable approaching $a$ ; it is not fixed at $a$ but represents a moving quantity, and if you set $x = a$ too early, you will skip the limit condition.
$a$	The target point where continuity is tested; it must be in the function's domain, and if you do not check that $f(a)$ is defined, you will overlook discontinuities from missing values.

When to use it

Check continuity when you need to confirm that a function behaves smoothly at a specific point, especially before applying theorems like the Intermediate Value Theorem.

Level

Usually taught in: Calculus I

Worked examples

1. Check continuity of a simple polynomial

Problem

f(x) = x^2 + 2x

continuous at

x = 1

$f(1) = 1^{2} + 2(1) = 3$
The function is defined at $x = 1$ and equals 3.
$\lim_{x \to 1} (x^{2} + 2x) = 1^{2} + 2(1) = 3$
The limit as $x$ approaches 1 exists and also equals 3 by direct substitution since polynomials are continuous.
$\lim_{x \to 1} f(x) = f(1)$
Since the limit and function value are equal, all three requirements for continuity at $x = 1$ are satisfied.

Answer: $f(x) \text{ is continuous at } x = 1$

Polynomials are continuous at every point in their domain because they are built from basic operations on the variable $x$ that preserve continuity. To verify, we check that the limit equals the function value, which holds for all polynomials by direct substitution.

2. Identify a jump discontinuity in a piecewise function

Problem

For

g(x) = \begin{cases} x + 1 & \text{if } x < 1 \\ 4 & \text{if } x \geq 1 \end{cases}

, is

g

continuous at

x = 1

$\lim_{x \to 1^{-}} g(x) = \lim_{x \to 1^{-}} (x + 1) = 1 + 1 = 2$
From the left, the function follows the first piece: $x + 1 = 2$ .
$\lim_{x \to 1^{+}} g(x) = \lim_{x \to 1^{+}} 4 = 4$
From the right, the function is constantly 4.
$\lim_{x \to 1^{-}} g(x) = 2 \neq 4 = \lim_{x \to 1^{+}} g(x)$
The one-sided limits are different, so the overall limit does not exist.
$\lim_{x \to 1} g(x) \text{ does not exist, so } g \text{ is not continuous at } x = 1$
Since the limit requirement fails, continuity cannot hold, and $g$ has a jump discontinuity at $x = 1$ .

Answer: $g(x) \text{ is not continuous at } x = 1$

When the left and right limits of a piecewise function are different at a transition point, the overall limit does not exist. A jump discontinuity occurs whenever a function 'jumps' from one value to another, and this violates the continuity definition.

3. Verify continuity in a real-world temperature measurement

Problem

A lab thermometer measures a cooling liquid with temperature

T(t) = \begin{cases} 100 - 5t & \text{if } 0 \leq t \leq 6 \\ 70 & \text{if } t > 6 \end{cases}

degrees Celsius, where

t

is minutes. Verify whether

T

is continuous at

t = 6

$\lim_{t \to 6^{-}} T(t) = \lim_{t \to 6^{-}} (100 - 5t) = 100 - 5(6) = 100 - 30 = 70$
As time approaches 6 minutes from below, the temperature is given by $100 - 5t = 70$ .
$\lim_{t \to 6^{+}} T(t) = \lim_{t \to 6^{+}} 70 = 70$
As time approaches 6 minutes from above, the temperature is constant at 70°C.
$T(6) = 100 - 5(6) = 70$
At exactly $t = 6$ , the temperature is 70°C.
$\lim_{t \to 6} T(t) = 70 = T(6)$
All three conditions for continuity are met: the limit exists, the function value exists, and they are equal.

Answer: $T(t) \text{ is continuous at } t = 6 \text{ minutes}$

In physical systems, temperature typically changes continuously without sudden jumps. This piecewise function models a cooling process where the liquid reaches equilibrium at 70°C by $t = 6$ . The continuity at $t = 6$ confirms that the transition from active cooling to equilibrium is smooth—the temperature reaches exactly 70°C at that moment, not above or below.

Common mistakes

Where Continuity usually goes wrong

The mistake	Why it is wrong	The fix
Checking only that the limit exists and concluding the function is continuous without verifying that $f(a)$ equals the limit.	The continuity definition requires all three conditions: $f(a)$ must be defined, the limit $\lim_{x \to a} f(x)$ must exist, and they must be equal.	Always verify the function value $f(a)$ and compare it to the limit; continuity fails if either is missing or if they differ.
Assuming a function is continuous at a transition point of a piecewise function just because the formula changes there.	A change in formula does not automatically mean discontinuity; continuity depends on whether the pieces 'match up' (the limit equals the function value), not on the formula itself.	Check one-sided limits and the function value at transition points regardless; they might agree even when the formula changes.
Concluding that a function is continuous just because the left and right limits are equal.	Even if $\lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x)$ , the function is only continuous if this common limit also equals $f(a)$ .	After finding that both one-sided limits are equal, verify that the limit value equals $f(a)$ to confirm continuity.

Tips and when to use something else

Check continuity at transition points of piecewise functions using one-sided limits; both must equal each other and match $f(a)$ .
Polynomials, exponential functions, trigonometric functions like sine and cosine, and rational functions (where defined) are continuous everywhere in their domain, so you can evaluate their limits by direct substitution.
If the left and right limits are different, the overall limit does not exist and continuity automatically fails—no need to check further.
When the limit exists but $f(a)$ is undefined or different, you have a removable discontinuity; investigate this case using the Removable Discontinuity method instead of concluding non-continuity outright.

Related formulas

Definition of a LimitDefinition of a Limit describes what value a function approaches as its input approaches a specific point—essential for all of calculus.One-Sided LimitsOne-sided limits let you examine how a function behaves as you approach a point from just the left or right, determining whether a two-sided limit exists.Removable DiscontinuityA removable discontinuity is a hole in a graph where the limit exists but the function value differs; it can be fixed by redefining that point.Intermediate Value TheoremIntermediate Value Theorem states that a continuous function on a closed interval must cross any value between its endpoints.

Frequently asked questions

What does it mean for a function to be continuous at a point?: A function $f$ is continuous at point $a$ when three conditions are satisfied: $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and they are equal. Geometrically, this means you can draw the function through that point without lifting your pencil.
How do you check if a piecewise function is continuous at a breakpoint?: Use one-sided limits. Calculate $\lim_{x \to a^{-}} f(x)$ using the left piece and $\lim_{x \to a^{+}} f(x)$ using the right piece. If both one-sided limits are equal and match $f(a)$ , the function is continuous at the breakpoint.
Is every polynomial continuous?: Yes. All polynomial functions are continuous everywhere in their domain because they are sums and products of the simple function $f(x) = x$ , which is continuous. This means you can always evaluate a polynomial's limit by direct substitution.
What is a removable discontinuity and how is it different from a jump discontinuity?: A removable discontinuity occurs when the limit exists but the function value is either undefined or different from the limit. A jump discontinuity occurs when the left and right limits exist but are not equal. Removable discontinuities can sometimes be 'fixed' by redefining the function value, whereas jump discontinuities cannot.

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Reviewed 2026-09-18