Problem 53 - Thomas Calculus

Question

Greatest integer in \(x\) Does any function differentiable on \((-\infty, \infty)\) have \(y=\operatorname{int} x,\) the greatest integer in \(x\) (see Figure \(2.55 ),\) as its derivative? Give reasons for your answer.

Step-by-Step Solution

Answer

No, because the greatest integer function is not continuous.

Step 1: Analyze the Greatest Integer Function

The function \(y = \operatorname{int} x\), also known as the floor function, maps any real number \(x\) to the greatest integer less than or equal to \(x\). This function is a step function, which means it jumps from one integer value to the next at every integer point.

Step 2: Consider the Properties of a Derivative

A function is differentiable on an interval if it is continuous and smooth (i.e., it has a defined tangent) at every point in that interval. The derivative of a function indicates its instantaneous rate of change at each point, which translates to the slope of the tangent to the curve at each point.

Step 3: Determine the Differentiability of Int(x)

Since \(y = \operatorname{int} x\) is a step function with jumps at every integer point, it is not continuous at those points. This means that the floor function itself is not differentiable where these jumps occur, as the derivative doesn't exist at these discontinuities.

Step 4: Investigate the Possibility of an Anti-Derivative

If a differentiable function exists with \(y = \operatorname{int} x\) as its derivative, then \(\operatorname{int} x\) must fulfill the conditions for being a derivative, particularly requiring continuity. Since \(\operatorname{int} x\) is not continuous due to its jump discontinuities, it cannot be the derivative of any differentiable function.

Key Concepts

Greatest Integer Function

The greatest integer function, also referred to as the floor function, is an interesting mathematical concept. It maps each real number to the largest integer less than or equal to that number. For example, for the input value of 2.7, the greatest integer function would output 2, since 2 is the largest integer that is less than or equal to 2.7.
This function is symbolized as \( y = \operatorname{int}(x) \) or \( y = \lfloor x \rfloor \). It is noteworthy that this function is not linear because it does not produce a continuous, straight line. Instead, it forms a "stairstep" pattern, moving horizontally until encountering the next integer, then moving vertically at that integer point.

• This abrupt change at integer boundaries makes the greatest integer function discontinuous at those points.
• These jumps or steps make it part of a group of functions known as **step functions**.

Step Function

A step function is a type of function that remains constant within certain intervals and jumps to new constant values at specific points. The greatest integer function is a classic example of a step function. These are defined as having piecewise constant values, meaning:

• They hold a certain value for an interval (part of their domain).
• They transition instantaneously to a new value at certain defined points.

The main characteristic of step functions is their discontinuity. Each time they transition from one step to another, the function is not continuous. In the graph of the greatest integer function, for example, you can see it step horizontally between integer values like 1, 2, 3, and so on, then jump vertically at those integer points. This characteristic makes step functions different from typical continuous functions, as their graphs often have these apparent gaps or jumps.

Continuous Functions

Continuous functions are at the heart of calculus and differentiate from step functions. A function is continuous if there are no breaks, jumps, or gaps in its graph. This means that you can draw the graph of a continuous function without lifting your pencil or pen from the paper. Mathematically, a function \( f(x) \) is continuous at a point \( a \) if:

• \( f(a) \) is defined (the function has a value at \( a \))
• The limit of \( f(x) \) as \( x \) approaches \( a \) exists
• The limit equals the function's value at that point, i.e., \( \lim_{{x \to a}} f(x) = f(a) \)

Step functions, such as the greatest integer function, are not continuous because of their inherent jumps at integer boundaries. These jumps mean the conditions for continuity are breached at those points, leading to discontinuities.

Derivative Properties

Differentiability refers to a function having a well-defined derivative at every point within its domain. For a function to be differentiable at a point, it must also be continuous at that point. Thus, if a function isn’t continuous somewhere, it can’t be differentiable there.
The derivative represents the slope of the tangent line to the curve of the function at any given point. If a function, like the greatest integer function, has jumps or is non-continuous at certain points, it cannot have a derivative there, indicating non-differentiability at those points.

• A function can have endpoints or corners and still not be differentiable there even if it's continuous.
• In the realm of calculus, step functions prove challenging since their discontinuous nature falls outside typical smooth curve assumptions.

Thus, since the greatest integer function isn't continuous at its integer points, it doesn't fit the necessary requirements to have a derivative at those points, making it not suitable as the derivative of any differentiable function.