What is Continuity?

🔗 Continuity — Drawing Without Lifting the Pencil

Part 1 of 4 — The intuition

Topics in This Part

🔑 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:

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

In short: the limit equals the value.

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

⚖️ Continuous vs. Not

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

Three-Part Test 🎯

Identify 🧮

For each statement, type yes or no:

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

2) Is $g(x) = 1/(x - 4)$ continuous at $x = 4$?

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

📜 Continuity Catalog

Part 2 of 4 — Which functions are continuous where?

Topics in This Part

🔑 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:

💡 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.

⚠️ 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 🔽

🧩 Continuity for Piecewise Functions

Part 3 of 4 — Make the pieces match up

Topics in This Part

🔑 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

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

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 $x = 2$: $2k + 3$.
• Set equal: $2k + 3 = 5 \Rightarrow k = 1$.

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 $x = 1$: $a(1) + b = 1^2 \Rightarrow a + b = 1$.
• At $x = 2$: $2^2 = 4(2) + a \Rightarrow 4 = 8 + a \Rightarrow a = -4$.
• From the first equation: $b = 5$.

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 $a$.

🌐 Continuity on an Interval & the IVT

Part 4 of 4 — Big-picture continuity

Topics in This Part

🔑 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 $(a, b)$.

$f$ 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 $[0, 1]$ ✅
• $f(0) = -1 < 0$
• $f(1) = 1 > 0$
• $0$ is strictly between $-1$ and $1$.

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

💡 IVT guarantees existence; it doesn't tell you exactly where the root is. On the exam, a calculator finds the value when one is needed.

IVT Application 🎯

Exit Check ✅