Piecewise continuous functions
Piecewise Continuous Functions
Piecewise continuous functions are functions that are continuous within intervals but may have a finite number of discontinuities at certain points. These functions can be defined by different expressions over different intervals. Understanding piecewise continuous functions is crucial for the study of definite integrals, as they often arise in practical applications.
Definition
A function $f(x)$ is said to be piecewise continuous on an interval $[a, b]$ if:
- The interval $[a, b]$ can be divided into a finite number of subintervals.
- Within each subinterval, the function is continuous.
- At the endpoints of these subintervals, the function may have jump discontinuities, finite discontinuities, or removable discontinuities.
Mathematically, if $[a, b]$ is divided into subintervals $[x_{i-1}, x_i]$ for $i = 1, 2, ..., n$ where $a = x_0 < x_1 < ... < x_n = b$, then $f(x)$ is continuous on $(x_{i-1}, x_i)$ for all $i$.
Properties
Piecewise continuous functions have several important properties:
- They can be integrated over the interval $[a, b]$.
- They may have a finite number of discontinuities.
- The limits of the function exist at the points of discontinuity.
Table of Differences and Important Points
| Property | Continuous Functions | Piecewise Continuous Functions |
|---|---|---|
| Definition | A function is continuous at a point if the limit as it approaches the point from both sides is equal to the function's value at that point. | A function is piecewise continuous if it is continuous within certain intervals and may have a finite number of discontinuities. |
| Discontinuities | None within the interval of interest. | A finite number of discontinuities are allowed. |
| Integrability | Always integrable over an interval where it is continuous. | Integrable over the interval, but special consideration may be needed at points of discontinuity. |
| Limits | The limit of the function exists and is equal to the function's value at every point in its domain. | Limits exist at points of discontinuity, but the function's value may not be defined, or may not equal the limit. |
Formulas
The definite integral of a piecewise continuous function $f(x)$ over an interval $[a, b]$ can be computed by breaking the integral into the sum of integrals over the subintervals:
$$ \int_{a}^{b} f(x) \, dx = \int_{a}^{x_1} f(x) \, dx + \int_{x_1}^{x_2} f(x) \, dx + \cdots + \int_{x_{n-1}}^{b} f(x) \, dx $$
Examples
Example 1: Piecewise Linear Function
Consider the function $f(x)$ defined as:
$$ f(x) = \begin{cases} x & \text{if } 0 \leq x < 1 \ 2 - x & \text{if } 1 \leq x \leq 2 \end{cases} $$
This function is piecewise continuous because it is defined by two continuous functions over two subintervals of $[0, 2]$. There is a discontinuity at $x = 1$, but it is a removable discontinuity since the limit from both sides is 1.
Example 2: Piecewise Function with Jump Discontinuity
Consider the function $g(x)$ defined as:
$$ g(x) = \begin{cases} x^2 & \text{if } -1 \leq x < 0 \ x + 1 & \text{if } 0 \leq x \leq 1 \end{cases} $$
Here, $g(x)$ is piecewise continuous with a jump discontinuity at $x = 0$. The left-hand limit as $x$ approaches 0 is 0, and the right-hand limit is 1. The function is continuous within the intervals $[-1, 0)$ and $(0, 1]$.
Example 3: Definite Integral of a Piecewise Function
Let's find the definite integral of the function $g(x)$ from the previous example over the interval $[-1, 1]$:
$$ \int_{-1}^{1} g(x) \, dx = \int_{-1}^{0} x^2 \, dx + \int_{0}^{1} (x + 1) \, dx $$
Computing these integrals separately, we get:
$$ \int_{-1}^{0} x^2 \, dx = \left[\frac{x^3}{3}\right]_{-1}^{0} = 0 - \left(-\frac{1}{3}\right) = \frac{1}{3} $$
$$ \int_{0}^{1} (x + 1) \, dx = \left[\frac{x^2}{2} + x\right]_{0}^{1} = \left(\frac{1}{2} + 1\right) - (0 + 0) = \frac{3}{2} $$
Adding these results together gives us the total integral:
$$ \int_{-1}^{1} g(x) \, dx = \frac{1}{3} + \frac{3}{2} = \frac{11}{6} $$
Piecewise continuous functions are essential in various fields, including engineering and physics, where they model systems that behave differently under different conditions. Understanding how to work with these functions is a key skill in applied mathematics.