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Piecewise Functions

Definition

A function that is defined by different analytical expressions in different parts of its domain is called a piecewise function.

Characteristics

Piecewise functions have the following characteristics:

Domain Partitioning: The domain is divided into several intervals
Different Expressions: Each interval corresponds to an expression
Breakpoint Handling: Continuity at interval boundaries needs special attention
Piecewise Analysis: Function properties need to be analyzed piecewise

Construction Methods

The construction of piecewise functions usually follows these steps:

Determine Domain: Clarify the function’s domain
Divide Intervals: Partition the domain into different intervals based on conditions
Choose Expressions: Select appropriate expressions for each interval
Handle Breakpoints: Ensure function definition and continuity at breakpoints

Common Examples

Absolute Value Function

f(x) = |x| = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases}

Sign Function

f(x) = \text{sgn}(x) = \begin{cases} 1, & x > 0 \\ 0, & x = 0 \\ -1, & x < 0 \end{cases}

Floor Function

f(x) = [x] = \text{the greatest integer less than or equal to } x

Notes

Breakpoint Definition: The value of piecewise functions at breakpoints must be explicit
Continuity: Piecewise functions may be discontinuous at certain points
Property Analysis: Properties of piecewise functions need piecewise discussion
Graph Drawing: When drawing piecewise function graphs, pay attention to segment connections

Property Analysis of Piecewise Functions

Continuity

The continuity of piecewise functions at breakpoints requires special attention:

If left and right limits are equal and equal to the function value, then continuous
Otherwise the function is discontinuous at that point

Differentiability

The differentiability of piecewise functions at breakpoints:

Check if left and right derivatives are equal
If equal, then differentiable at that point
Otherwise not differentiable at that point

Exercises

Exercise 1

Construct a piecewise function $f(x)$ such that:

When $x \geq 0$ , $f(x) = x^2$
When $x < 0$ , $f(x) = -x$

Reference Answer

piecewise functionconstruction

Approach: Construct the piecewise function according to the given conditions.

Steps:

According to the conditions, use $x^2$ when $x \geq 0$
Use $-x$ when $x < 0$
At the breakpoint $x = 0$ , $f(0) = 0^2 = 0$

Answer:

f(x) = \begin{cases} x^2, & x \geq 0 \\ -x, & x < 0 \end{cases}

Exercise 2

Determine the continuity of the piecewise function $f(x) = \begin{cases} x^2, & x \geq 0 \\ -x, & x < 0 \end{cases}$ at $x = 0$ .

Reference Answer

piecewise functioncontinuity

Approach: Need to check the left limit, right limit, and function value at $x = 0$ .

Steps:

Calculate left limit: $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (-x) = 0$
Calculate right limit: $\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x^2 = 0$
Function value: $f(0) = 0^2 = 0$
Since left limit, right limit, and function value are all equal, the function is continuous at $x = 0$ .

Answer:

The function is continuous at $x = 0$ .

Exercise 3

Construct a piecewise function $f(x)$ such that:

When $x > 1$ , $f(x) = \frac{1}{x-1}$
When $x \leq 1$ , $f(x) = x^2$

Reference Answer

piecewise functionconstructioncontinuity

Approach: Construct the piecewise function according to the given conditions.

Steps:

According to the conditions, use $\frac{1}{x-1}$ when $x > 1$
Use $x^2$ when $x \leq 1$
At the breakpoint $x = 1$ , $f(1) = 1^2 = 1$

Answer:

f(x) = \begin{cases} \frac{1}{x-1}, & x > 1 \\ x^2, & x \leq 1 \end{cases}

Note: This function is discontinuous at $x = 1$ because the left limit is $1$ and the right limit is $+\infty$ .

Summary

Symbols Appearing in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$f(x)$	Mathematical symbol	f of x	Function notation, representing a function with $x$ as the independent variable
$\\|x\\|$	Mathematical Symbol	absolute value of x	Represents the absolute value of $x$
$\text{sgn}(x)$	Mathematical symbol	sign of x	Sign function
$[x]$	Mathematical symbol	floor of x	Floor function, the greatest integer less than or equal to $x$
$\lim$	Mathematical symbol	limit	Limit symbol
$x \to 0^-$	Mathematical symbol	x approaches zero from left	$x$ approaches 0 from the left
$x \to 0^+$	Mathematical symbol	x approaches zero from right	$x$ approaches 0 from the right

Inverse Functions

Implicit Functions

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Piecewise Functions

Definition

Characteristics

Construction Methods

Common Examples

Absolute Value Function

Sign Function

Floor Function

Notes

Property Analysis of Piecewise Functions

Continuity

Differentiability

Exercises

Exercise 1

Exercise 2

Exercise 3

Summary

Symbols Appearing in This Article

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