Help me learn the key concepts of piecewise functions in math

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A piecewise function is a mathematical function that is defined by multiple sub-functions, each applying to a different interval of the independent variable's domain. Unlike regular functions with a single formula, piecewise functions use different rules for different parts of their domain. This example shows a function with three pieces: a linear function for x less than negative one, a constant function for x between negative one and one, and another linear function for x greater than or equal to one. Each piece of a piecewise function is a standard mathematical function. The first piece might be quadratic, like one-half x squared. The second piece could be linear, like negative x plus one. The third piece might be constant, like the value two. Each piece applies to its own specific domain interval, defined by inequalities. Notice how the endpoints use open or closed circles to show whether that point is included in each piece's domain. To evaluate a piecewise function for a given input, follow these steps: First, identify which interval the input falls into by checking the domain conditions. Second, use the corresponding piece formula for that interval. Third, calculate the result. For example, to find f of negative 1.5, we see that negative 1.5 is less than negative 1, so we use the first piece: x plus 2. This gives us negative 1.5 plus 2, which equals 0.5. For f of 2, since 2 is greater than or equal to 1, we use the third piece: x. So f of 2 equals 2. Piecewise functions can be continuous or discontinuous at boundary points where pieces meet. A function is continuous at a point if the left-hand limit equals the right-hand limit, and both equal the function value at that point. The blue function shown is continuous at x equals zero because both pieces meet at the same point. However, piecewise functions can also be discontinuous, creating jumps, holes, or breaks in the graph. The red function demonstrates a jump discontinuity where the pieces don't connect smoothly. Piecewise functions are essential for modeling real-world situations where rules change at specific points. Common applications include tax brackets with different rates for income levels, shipping costs that vary by weight tiers, utility bills with usage-based pricing, and parking fees with time intervals. This example shows a parking garage fee structure: five dollars flat rate for up to two hours, then three dollars per hour for the next four hours, and two dollars per hour after six hours. These functions help businesses and organizations create fair, structured pricing systems.