This is a question from Differential calculus based on greatest integer function. Please provide answer as soon as possible.

Let's dive into the concept of the greatest integer function, often denoted as ⌊x⌋, which represents the largest integer less than or equal to x. This function is particularly interesting in differential calculus because it is a piecewise constant function, meaning it has constant values over certain intervals but jumps at integer points.

Understanding the Greatest Integer Function

The greatest integer function can be defined mathematically as:

• If x is an integer, then ⌊x⌋ = x.
• If x is not an integer, then ⌊x⌋ is the integer part of x, which is the largest integer less than x.

For example:

• ⌊3.7⌋ = 3
• ⌊-2.3⌋ = -3
• ⌊5⌋ = 5

Behavior of the Function

Graphically, the greatest integer function appears as a series of horizontal steps. Each step corresponds to an interval between two consecutive integers. For instance, from 0 to 1, the function value is 0; from 1 to 2, the function value is 1, and so on. At each integer point, the function jumps up to the next integer.

Differentiability of the Function

Now, when we consider the differentiability of the greatest integer function, we find that it is not differentiable at integer points. This is because the function has a discontinuity at these points; it jumps from one integer value to another. However, it is differentiable everywhere else in its domain, where it is constant.

Formal Explanation of Differentiability

To determine if a function is differentiable at a point, we check if the limit of the difference quotient exists:

For a function f(x), the derivative at a point a is given by:

f'(a) = lim (h → 0) [(f(a + h) - f(a)) / h]

For the greatest integer function at an integer point, say a = n:

• For h approaching 0 from the right (h > 0), f(n + h) = n.
• For h approaching 0 from the left (h < 0), f(n + h) = n - 1.

Thus, we have:

f'(n) = lim (h → 0) [(n - n) / h] = 0 (from the right)

f'(n) = lim (h → 0) [(n - 1 - n) / h] = -∞ (from the left)

Since the left-hand limit and right-hand limit do not match, the derivative does not exist at integer points.

Conclusion on the Greatest Integer Function

In summary, the greatest integer function is a fascinating example in differential calculus. It is piecewise constant, making it differentiable everywhere except at integer points where it is discontinuous. Understanding this behavior is crucial for analyzing functions that exhibit similar characteristics in calculus.