Chapter 5 · Continuity and Differentiability

Example 6: The identity function $f(x)=x$ is continuous everywhere

ExamplesContinuous functions · Section 5.2

The question

Prove that the identity function on real numbers given by $f(x) = x$ is continuous at every real number.

Exam answer

Let $c$ be any real number. Then $f(c) = c$ ⁠.

\text{LHL} = \lim_{x\to c^-} x = c

\text{RHL} = \lim_{x\to c^+} x = c

Since $\text{LHL} = \text{RHL} = f(c)$ ⁠, $f$ is continuous at $x = c$ ⁠.

As $c$ is arbitrary, $f(x) = x$ is continuous at every real number. Hence proved.

Want the reason behind every step, the basics and the checks?

The one idea this problem is built on

The identity function returns its input unchanged, so the distance between $f(x)$ and $f(c)$ is exactly the distance between $x$ and $c$ . Bringing $x$ close to $c$ therefore brings $f(x)$ equally close to $f(c)$ ⁠, at every point $c$ ⁠. The proof is this observation, carried out at an arbitrary point.

Given, to prove, and the plan

Given: $f(x) = x$ for every real $x$ ⁠.
To prove: $f$ is continuous at every real number.
Plan: Take an arbitrary real $c$ ⁠. Find $f(c)$ ⁠, the left-hand limit and the right-hand limit at $c$ ⁠, and show they are equal (Definition 1).

Basics you need

1The identity function

$f(x) = x$ sends every number to itself: $f(3) = 3$ ⁠, $f(-0.5) = -0.5$ ⁠. Its graph is the line $y = x$ through the origin at 45°. Basics: functions

2What a limit is

$\lim_{x\to c} f(x) = L$ means $f(x)$ can be made as close to $L$ as we like by taking $x$ close enough to $c$ ⁠, $x \ne c$ ⁠. Basics: what a limit is

3One-sided limits

$x\to c^-$ uses only $x < c$ ⁠, and $x\to c^+$ only $x > c$ ⁠. The limit exists when both exist and agree. Basics: left-hand and right-hand limits

4Definition 1

For $c$ in the domain, $f$ is continuous at $c$ iff the left-hand limit, the right-hand limit and $f(c)$ exist and are equal. Lesson: continuity at a point

Solution

Fix an arbitrary point and find $f(c)$

$f(x) = x$ is defined for every real $x$ ⁠, so the domain is $\R$ ⁠. Let $c$ be any real number. Then
$f(c) = c.$
"Continuous at every real number" is a "for every" statement, so we argue at a point $c$ about which nothing is assumed beyond being real. Continuity at $c$ is only meaningful because $c$ is in the domain.

The identity function's output equals its input. Basics: functions
Measure how far $f(x)$ is from $f(c)$

For every real $x$ ⁠,
$|f(x) - f(c)| = |x - c|.$
So if $x$ is within 0.01 of $c$ ⁠, then $f(x)$ is within 0.01 of $c$ ⁠; if $x$ is within 0.000001 of $c$ ⁠, so is $f(x)$ ⁠. Any demanded closeness is met by taking $x$ just as close to $c$ ⁠, from either side.

This is the meaning of a limit (Basic 2) applied directly. It is also the honest reason behind the textbook's line $\lim_{x\to c} x = c$ ⁠: that line is not a calculation but a direct consequence of $f(x) - c = x - c$ ⁠.

$|x - c|$ is the distance between $x$ and $c$ on the number line. Basics: what a limit is
The two one-sided limits

Step 2 holds for $x < c$ and for $x > c$ ⁠, so
$\begin{aligned} \lim_{x\to c^-} f(x) &= \lim_{x\to c^-} x = c, \\ \lim_{x\to c^+} f(x) &= \lim_{x\to c^+} x = c. \end{aligned}$
For example, with $c = 3$ ⁠:

$x$ $f(x)$ side of 3

$2.9$ $2.9$ left

$2.99$ $2.99$ left

$3.01$ $3.01$ right

$3.1$ $3.1$ right

The rule $x$ is the same on both sides of $c$ ⁠, so both one-sided limits come out the same way. Since they agree, $\lim_{x\to c} f(x) = c$ ⁠.
Compare and conclude
$\begin{aligned} \lim_{x\to c^-} f(x) &= c, \\ \lim_{x\to c^+} f(x) &= c, \\ f(c) &= c. \end{aligned}$
All three agree, so $\lim_{x\to c} f(x) = c = f(c)$ and $f$ is continuous at $c$ ⁠. Because $c$ was arbitrary, the identity function is continuous at every real number.

Definition 1 is satisfied at the arbitrary point $c$ ⁠, which proves it at every real number. So $f$ is a continuous function (Definition 2).

Lesson: continuity at a point

$x$	$f(x)$	side of 3
$2.9$	$2.9$	left
$2.99$	$2.99$	left
$3.01$	$3.01$	right
$3.1$	$3.1$	right

The identity function

y = x

(same scale on both axes). At any point

c

⁠, the graph arrives at height

c

from both sides, and

(c, c)

is on the graph. The dashed guides show that the height equals the input.

Result

For every real $c$ ⁠, $\lim_{x\to c} f(x) = c = f(c)$ ⁠. Hence the identity function $f(x) = x$ is continuous at every real number, as the textbook proves.

Check it with numbers

At $c = -1.5$ ⁠: $f(-1.5004) = -1.5004$ and $f(-1.4996) = -1.4996$ ⁠, each $0.0004$ from $f(-1.5) = -1.5$ ⁠, exactly the distance of $x$ from $-1.5$ ⁠.

Common mistakes

Proving it at one point only. "Continuous at every real number" needs an arbitrary $c$ ⁠, not $c = 0$ or $c = 1$ ⁠.
Treating $\lim_{x\to c} x = c$ as obvious without reason. It is true because $|f(x) - c| = |x - c|$ ⁠; that is worth saying once.
Mixing up $x$ and $c$ ⁠. $c$ is the fixed point being tested; $x$ is the moving input that approaches it.

What to take away

The identity function is continuous everywhere because it does not change distances.
Constants (Example 5) and the identity function are the two building blocks. Sums, products and quotients of continuous functions are continuous (Theorem 1, Section 5.2.1), so every polynomial $a_0 + a_1x + \dots + a_nx^n$ is continuous (Example 14).

Every fact used, and where it comes from

Fact	Used in	Source
Identity function: $f(x) = x$	Steps 1–3	Definition (Basics: functions)
$\|f(x) - f(c)\| = \|x - c\|$	Step 2	Arithmetic
Meaning of a limit; $\lim_{x\to c} x = c$	Steps 2, 3	Limits, Class XI (Limits and Derivatives) (Basics: what a limit is)
Continuous at $c$ iff LHL = RHL = $f(c)$	Step 4	Definition 1, Section 5.2 (Lesson: continuity at a point)
Continuous function	Step 4	Definition 2, Section 5.2
Sums and products of continuous functions are continuous	Take-away 2	Theorem 1, Section 5.2.1 (Lesson: algebra of continuous functions)

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Example 6: The identity function f(x)=xf(x)=xf(x)=x is continuous everywhere

Basics you need

Solution

Fix an arbitrary point and find f(c)f(c)f(c)

Measure how far f(x)f(x)f(x) is from f(c)f(c)f(c)

The two one-sided limits

Compare and conclude

Every fact used, and where it comes from

Example 6: The identity function $f(x)=x$ is continuous everywhere

Fix an arbitrary point and find $f(c)$

Measure how far $f(x)$ is from $f(c)$