Continuity

Warm-up

Three quick questions from earlier lessons.

Discover

You can draw the graph of a continuous function without lifting your pencil. Which one can you draw that way near $x = 0$ ?

$y = |x|$
$y = \dfrac{1}{x}$

Key idea

Continuity at a point means: no hole, no jump, no asymptote there.

f

is continuous at $a$ when

f(a)

exists,

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

exists, and they are equal. Polynomials,

\sin

,

\cos

,

e^x

and

|x|

are continuous everywhere; rational functions are continuous wherever they are defined.

Worked example

For which $k$ is $f(x) = \begin{cases} x + k & \text{if } x < 2 \\ x^2 & \text{if } x \ge 2 \end{cases}$ continuous?

f(2) = 4

f(2) uses the second piece.

\displaystyle\lim_{x \to 2^-} f(x) = 2 + k

The left limit uses the first piece.

2 + k = 4

, so

k = 2

They must be equal.

Your turn, guided

Your turn: $g(x) = \begin{cases} 3x & \text{if } x < 1 \\ x + m & \text{if } x \ge 1 \end{cases}$ .

3 = 1 + m

Left limit 3, value 1 + m.

m = 2

Solve for m.

Practice ★

For which $k$ is $f(x) = \begin{cases} x^2 + k & \text{if } x < 3 \\ 4x & \text{if } x \ge 3 \end{cases}$ continuous everywhere?
For which $k$ is $f(x) = \begin{cases} x^2 + k & \text{if } x < 3 \\ 8x & \text{if } x \ge 3 \end{cases}$ continuous everywhere?
For which $k$ is $f(x) = \begin{cases} x^2 + k & \text{if } x < 3 \\ 7x & \text{if } x \ge 3 \end{cases}$ continuous everywhere?

Watch out!

Practice ★★

Is $f(x) = \dfrac{x^2 - 25}{x - 5}$ continuous at $x = 5$ ?
1. No: $f(5)$ is not defined
2. Yes: the limit exists, it is $10$
3. No: the limit does not exist
Is $f(x) = \dfrac{x^2 - 81}{x - 9}$ continuous at $x = 9$ ?
1. No: $f(9)$ is not defined
2. Yes: the limit exists, it is $18$
3. No: the limit does not exist

Which functions are continuous for every real $x$ ? Tick all of them.
1. $x^3 - 2x + 1$
2. $\sin x$
3. $|x|$
4. $\dfrac{1}{x}$
5. $\tan x$

Exit ticket

Three questions to check you have got it.

For which $k$ is $f(x) = \begin{cases} x^2 + k & \text{if } x < 3 \\ 4x & \text{if } x \ge 3 \end{cases}$ continuous everywhere?
For which $k$ is $f(x) = \begin{cases} x^2 + k & \text{if } x < 3 \\ 8x & \text{if } x \ge 3 \end{cases}$ continuous everywhere?
For which $k$ is $f(x) = \begin{cases} x^2 + k & \text{if } x < 3 \\ 7x & \text{if } x \ge 3 \end{cases}$ continuous everywhere?

Summary card

Continuity

$f(a)$ exists.
$\displaystyle\lim_{x \to a} f(x)$ exists.
They are equal.

Watch out: a limit that exists is not enough; $f(a)$ must match it.

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