🎯⭐ INTERACTIVE LESSON

What is Continuity?

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What is Continuity? - Complete Interactive Lesson

Part 1: Continuity Definition

🔗 Continuity — Drawing Without Lifting the Pencil

Part 1 of 4 — The intuition

Topics in This Part

Section
Intuitive Definition
🔑 Limit = Value
Continuous vs. Not

🔑 Why this matters: Continuity is the bridge from limits to derivatives. It's the most heavily tested concept in Unit 1.

✏️ Intuitive Definition

A function $f$ is continuous on an interval if you can draw its graph without lifting your pencil.

Visually, this means no jumps, no holes, no vertical asymptotes.

This is great intuition but not quite a definition. The precise version uses limits.

🔑 Formal Definition: Continuous at $a$

A function $f$ is continuous at $x = a$ if all three conditions hold:

$f(a)$ is defined.
$\lim_{x}$ exists (both one-sided limits agree on a finite value).

⚖️ Continuous vs. Not

Function	Continuous at $a$ ?	Why
$f(x) = x^2$ at $a = 3$

Three-Part Test 🎯

Identify 🧮

For each statement, type yes or no:

1) Is $f(x) = x^3 - 5x$ continuous at $x = 2$ ?

2) Is continuous at ?

Part 2: Continuity Catalog

📜 Continuity Catalog

Part 2 of 4 — Which functions are continuous where?

Topics in This Part

Section
🔑 The Continuous-Everywhere List
Continuity on Restricted Domains
Combinations Preserve Continuity

🔑 Why this matters: Knowing which families are continuous saves you from re-checking the three conditions every time.

🔑 Continuous Everywhere

These functions are continuous at every real number:

Family	Examples
Polynomials	$x^2 + 1$ , , any constant

Part 3: Piecewise Continuity

🧩 Continuity for Piecewise Functions

Part 3 of 4 — Make the pieces match up

Topics in This Part

Section
The Boundary Test
🔑 Solving for an Unknown Constant
Worked Examples

🔑 Why this matters: AP loves to ask "what value of $k$ makes this piecewise function continuous?" — a routine application of the three-condition test.

📍 The Boundary Test

For a piecewise function defined as

$f (x) = {\begin{cases} g (x), & x < a \\ h (x), \end{cases}$

Part 4: Continuity on Intervals & IVT

🌐 Continuity on an Interval & the IVT

Part 4 of 4 — Big-picture continuity

Topics in This Part

Section
Continuous on an Open vs. Closed Interval
🔑 The Intermediate Value Theorem
Applying IVT

🔑 Why this matters: Continuity on an interval enables the IVT — a foundational existence theorem you'll use in later units (and on the AP exam).

📏 Continuous on an Interval

$f$ is continuous on an open interval $(a, b)$ if it's continuous at every point in .

lim_{x \to a} f (x)

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

In short: the limit equals the value.

💡 If any one of the three conditions fails, $f$ is discontinuous at $a$ .

Most "common" functions (polynomials, sin, cos, exp) are continuous everywhere they're defined.

g (x) = 1/ (x - 4)

3) Is $h(x) = \sin x$ continuous at $x = \pi$ ?

💡 Plug-in works for any of these. No need to think.

🚧 Continuous on a Restricted Domain

Some functions are continuous only where they're defined — and they aren't defined everywhere.

Function	Continuous on
$\ln x$	$(0, \infty)$
$\sqrt{x}$	$[0, \infty)$
$\tan x$	All $x \ne \pi/2 + k\pi$
Rational $p(x)/q(x)$	All $x$ where $q(x) \ne 0$

⚠️ At the edge of the domain, you might have a one-sided continuity (e.g., $\sqrt{x}$ is continuous from the right at $x = 0$ ).

🔄 Combinations Preserve Continuity

If $f$ and $g$ are continuous at $a$ , then so are:

$f + g$ and $f - g$
$f \cdot g$
$f / g$ if $g(a) \ne 0$
$f \circ g$ (composition), provided $g$ is continuous at $a$ and $f$ is continuous at $g(a)$

So combinations of "nice" functions stay nice.

Example. $h(x) = \sin(e^x + x^2)$ — built from continuous pieces, continuous everywhere.

Catalog Test 🎯

Decide Continuity 🔽

f (x) = {g (x), h (x), x < a x \geq a

to be continuous at the boundary $x = a$ , we need:

$\lim_{x \to a^-} g(x) \;=\; \lim_{x \to a^+} h(x) \;=\; f(a) = h(a).$

In words: left piece and right piece must meet at the same value at $a$ .

💡 Inside each piece (away from the boundary), the function is just $g$ or $h$ — already continuous if those are continuous.

🔑 Solving for an Unknown Constant

Classic AP setup: find the value of $k$ that makes $f$ continuous.

Worked example. Find $k$ such that $f(x) = \begin{cases} x^2 + 1, & x \le 2 \\ kx + 3, & x > 2 \end{cases}$ is continuous at $x = 2$ .

Left piece at $x = 2$ : $4 + 1 = 5$ .
Right piece limit at $x = 2$ : $2k + 3$ .

Answer: $k = 1$ .

🎯 Two Unknowns

Sometimes both pieces have an unknown — then you need a system.

Example. Find $a, b$ so that $f(x) = \begin{cases} ax + b, & x < 1 \\ x^2, & 1 \le x \le 2 \\ 4x + a, & x > 2 \end{cases}$ is continuous everywhere.

At $x = 1$ : $a(1) + b = 1^2 \Rightarrow a + b = 1$ .

Answer: $a = -4$ , $b = 5$ .

Boundary Practice 🎯

Solve 🧮

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

2) For $g(x) = \begin{cases} ax + 5, & x < 2 \\ x^3, & x \ge 2 \end{cases}$ , find .

$f$ is continuous on a closed interval $[a, b]$ if it's continuous on $(a, b)$ AND right-continuous at $a$ AND left-continuous at $b$ .

The endpoint conditions:

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

💡 Most "nice" functions (polynomials, sin, cos, $e^x$ ) are continuous on every interval.

🔑 The Intermediate Value Theorem (IVT)

IVT. If $f$ is continuous on $[a, b]$ and $N$ is any value strictly between $f(a)$ and $f(b)$ , then there exists at least one $c \in (a, b)$ such that $f(c) = N$ .

In words: a continuous function on a closed interval takes on every value between its endpoint values.

💡 The IVT is an existence theorem — it tells you a $c$ exists, but not where it is or how many.

🔍 Applying IVT — Root-Finding

Worked example. Show $f(x) = x^3 + x - 1$ has a root on $[0, 1]$ .

$f$ is a polynomial → continuous on $[0, 1]$ ✅
$f(0) = -1 < 0$
$f (1) =$

By IVT, there exists $c \in (0, 1)$ with $f(c) = 0$ . A root exists in $(0, 1)$ .

💡 IVT guarantees existence; doesn't tell you the root's location. You'd use other methods (bisection, Newton's method) to find it.

IVT Application 🎯

Set equal:

2k + 3 = 5 \Rightarrow k = 1

x = 2

2^2 = 4(2) + a \Rightarrow 4 = 8 + a \Rightarrow a = -4

From the first equation:

b = 5

0

is strictly between

-1

and

1

What is Continuity?

What is Continuity? - Complete Interactive Lesson

Part 1: Continuity Definition

🔗 Continuity — Drawing Without Lifting the Pencil

Topics in This Part

✏️ Intuitive Definition

🔑 Formal Definition: Continuous at aaa

⚖️ Continuous vs. Not

Part 2: Continuity Catalog

📜 Continuity Catalog

Topics in This Part

🔑 Continuous Everywhere

Part 3: Piecewise Continuity

🧩 Continuity for Piecewise Functions

Topics in This Part

📍 The Boundary Test

Part 4: Continuity on Intervals & IVT

🌐 Continuity on an Interval & the IVT

Topics in This Part

📏 Continuous on an Interval

🚧 Continuous on a Restricted Domain

🔄 Combinations Preserve Continuity

🔑 Solving for an Unknown Constant

🎯 Two Unknowns

🔑 The Intermediate Value Theorem (IVT)

🔍 Applying IVT — Root-Finding

🔑 Formal Definition: Continuous at $a$