Piecewise-defined functions are functions defined by multiple sub-functors that each apply to a different interval within the domain, also called piecewise functions, hybrid functions, or definitions by cases. Rather than being a characteristic of the function, a piecewise definition is a way of expressing the function.

There is a different but related notion of a property holding piecewise in a function, which is used when the domain can be divided into intervals on which the property holds. It is important to note that this is not a property of the notion above, but a property of the function itself. We will use the example of a piecewise linear function (which happens to be a continuous function as well).

**Defined Piecewise-Graph Functions**

There are times when we come across a function that requires more than one formula in order to obtain the desired results. For example, in the toolkit functions, we introduced the absolute value function f(x)=∣x∣. When the domain is made up of all real numbers, and the range includes all values greater than or equal to 0, the absolute value is the magnitude, or modulus, of any real number. The distance on the number line from 0 is the absolute value. It is essential that the output exceed or equal 0 in each of these definitions.

If we enter 0, or a positive number, the output is the same as the input.

- f(x)=x if x≥0

Inputs with negative values result in the opposite output.

- f(x)=−x if x<0

The absolute value function can be thought of as a piecewise function given that it requires two different processes or pieces to function. Essentially, a piecewise function refers to a function that uses more than one formula to define the output over a set of different pieces of the domain.

**Continuity And Differentiability of Piecewise Functions**

The following conditions must be met for a piecewise function to be continuous on a given interval in its domain:

- Functions of its constituent parts are continuous over the intervals (subdomains);
- At each endpoint of the subdomains within that interval, there is no discontinuity.
- For example, the pictured function is piecewise continuous within each subdomain, but not continuously throughout the entire domain due to the discontinuity at x
_{o}. This position uses the value of the right function piece as indicated by the filled circle.

The following conditions have to be met in addition to those pertaining to continuity for a piecewise function to be differentiable in its domain:

- Each of its constituent functions can be differentiated on the corresponding open intervals,
- there are one-sided derivatives at each interval endpoint,
- The one-sided derivatives of two neighboring subintervals coincide when two subintervals touch.

**Specific Types of Mathematical Functions**

The same is true for career paths, as there are many different types of functions. A piecewise function is an example of a mathematical function. There are different parts to a piecewise function. The piecewise function has many parts, each of which has a specific job it performs when the conditions are met.

**For example:**

- f(x) = x – 2, x < 3
- f(x) = (x – 1)2, x ≥ 3

If this function is invoked with a <3 input, it behaves differently from when it is invoked with a ≥ 3. The absolute value function is the most common piecewise function.

**Examples of Piecewise Functions**

Here are some more examples of piecewise functions that you might find useful. This shows that they do not have to be restricted to only two equations, they can have a lot of parts as well.

**For example:**

f(x) = 4 – x, x ≤ -4

f(x) = (x + 2)/2, x > -4

Orhow about:

f(x) = -3, x > 1

f(x) = -2x, x = 1

f(x) = x/2, x < 1

And how about this one:

f(x) = -x, x < 0

f(x) = x2, 0 ≤ x < 4

f(x) = x – 5, 4 ≤ x < 6

f(x) = x/2, x ≥ 6