Explain Signum function , Greatest integer function and Fractional part function

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The signum function, also known as the sign function, returns the sign of a real number. It outputs 1 for positive numbers, 0 for zero, and negative 1 for negative numbers. The graph shows three distinct parts: a horizontal line at negative 1 for negative x values, a point at the origin for x equals zero, and a horizontal line at positive 1 for positive x values. The greatest integer function, also called the floor function, returns the largest integer that is less than or equal to the input. For example, the floor of 3.7 is 3, and the floor of negative 2.3 is negative 3. The graph resembles a staircase with horizontal steps, where each step represents a constant integer value over an interval. The fractional part function returns the non-integer part of a real number. It is defined as x minus the floor of x. The range is always between 0 inclusive and 1 exclusive. For example, the fractional part of 3.7 is 0.7, and surprisingly, the fractional part of negative 2.3 is also 0.7. The graph creates a sawtooth pattern that repeats every unit interval. These three functions are closely related. The key relationship is that any real number x equals its floor value plus its fractional part. The signum function has a range of negative 1, 0, and 1. The floor function creates a step pattern, while the fractional part function creates a sawtooth pattern with range from 0 to 1. All three are examples of piecewise functions with distinct behaviors in different intervals.