Greatest Integer function is both differentiable and integrable

[ekomm]

Post below split from 2011 thread.

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While I was on the net I came across the question on whether Greatest Integer Function is differentiable. The response from someone was that this function is not differentiable in the sense that there is no change of value in the dependent variable while the dependent variable is increasing: putting it in my own words. The truth is that the Greatest Integer function can both be differentiated as well as integrated. This is my personal observation based on mathematical reasoning and intuition. I did not go through everything others say on this.

Consider, the function y = [x]² between the range 1 to 10. Between 0 and 0.99.. the greatest integer is 0. Between 1 and 1.999.. the greatest integer is 1. between 2 and 2.99.. the greatest integer is 4, and so on in between the respective intervals. You will notice that in between the intervals the values do not change except at the boundaries. For example, in the first interval the dependent variable y remains as 0 despite the change in x from 0 to 0.99..

Since the values of the dependent variable changes only at the boundaries ( in this case there are jumps from 0 to 1, and from 1 to 4, etc) it means that the greatest integer function can only be differentiated within respective intervals. Thus it is a piece-wise differentiation since the function outputs changes only at the boundaries as the independent variable changes integer-wise.

Based on this, the derivative of y=[x]² is dy/dx = 2[x] . Between 0 and 0.99.. the derivative is 0. Between 1 and 1.999.. the derivative is 2. Between 2 and 2.999.. the derivative is 8. The rate of change is determined only at the boundaries and not within their respective intervals like other functions. the same applies to integration.

In integration, you will have to sum up all the pieces in between the intervals. This is my proposition.

 

[daon2]

Post below split from 2011 thread.

---

While I was on the net I came across the question on whether Greatest Integer Function is differentiable. The response from someone was that this function is not differentiable in the sense that there is no change of value in the dependent variable while the dependent variable is increasing: putting it in my own words. The truth is that the Greatest Integer function can both be differentiated as well as integrated. This is my personal observation based on mathematical reasoning and intuition. I did not go through everything others say on this.

Consider, the function y = [x]² between the range 1 to 10. Between 0 and 0.99.. the greatest integer is 0. Between 1 and 1.999.. the greatest integer is 1. between 2 and 2.99.. the greatest integer is 4, and so on in between the respective intervals. You will notice that in between the intervals the values do not change except at the boundaries. For example, in the first interval the dependent variable y remains as 0 despite the change in x from 0 to 0.99..


Since the values of the dependent variable changes only at the boundaries ( in this case there are jumps from 0 to 1, and from 1 to 4, etc) it means that the greatest integer function can only be differentiated within respective intervals. Thus it is a piece-wise differentiation since the function outputs changes only at the boundaries as the independent variable changes integer-wise.

Based on this, the derivative of y=[x]² is dy/dx = 2[x] . Between 0 and 0.99.. the derivative is 0. Between 1 and 1.999.. the derivative is 2. Between 2 and 2.999.. the derivative is 8. The rate of change is determined only at the boundaries and not within their respective intervals like other functions. the same applies to integration.

In integration, you will have to sum up all the pieces in between the intervals. This is my proposition.

When someone states a function is "not differentiable" they specifically mean " on its domain". For example f(x)=1/x is differentiable, even though it is not defined everywhere on R. A tangent line at every point on its graph can be appropriately determined. Functions like g(x)=[x] do not have tangent lines at its discontinuities, even though it is defined at them.

Integration is very different. Many more functions are integrable than differentiable. Discontinuity is not an immediate problem, though closed forms are much more of a problem.

Also (when defined), d/dx [x]^2 is 0, not 2[x].

 

[JeffM]

When someone states a function is "not differentiable" they specifically mean " on its domain". For example f(x)=1/x is differentiable, even though it is not defined everywhere on R. A tangent line at every point on its graph can be appropriately determined. Functions like g(x)=[x] do not have tangent lines at its discontinuities, even though it is defined at them.

Integration is very different. Many more functions are integrable than differentiable. Discontinuity is not an immediate problem, though closed forms are much more of a problem.

Also (when defined), d/dx [x]^2 is 0, not 2[x].

Perhaps it is me, but I find the above to be very obscure in relation to the question asked. Here is what I think was intended.

One usage of "differentiable" means "differentiable at any point in the domain." For example, we say that 1/x is a differentiable function even though the function is not defined at x = 0 because 0 is not part of the domain. Therefore, a function is "not differentiable" under that usage if there is at least one point in the function's domain at which the derivative does not exist.

{Personally I am not sure that usage is universal. I believe I have heard knowledgeable people say "the absolute value function is differentiable except at x = 0" or "the absolute value function is not differentiable at x = 0." That may be informal usage. Informal or not, that usage seems to apply only when the number of exceptions is quite small. I do not recollect ever hearing anyone say that a function was differentiable except at an infinite number of points in its domain.}

I do not see that it would be offensive to say that

\(\displaystyle f(x) = \lfloor x \rfloor \text { is differentiable in the interval } (n,\ n + 1) \text { where } n \in \mathbb Z.\)

But that is not the same as saying without explicit qualification that the floor function is differentiable.

Furthermore the derivative of that function is zero. Consequently, by the chain rule,

\(\displaystyle g(x) = \{f(x)\}^2 \implies g'(x) = 0.\)

And finally the OP in his text seems to be thinking about \(\displaystyle \lfloor x^2 \rfloor\),

which is is not differentiable unless we restrict the domain to (- 1, 0) or (0, 1).

Ahh well. I have probably misunderstood the OP and daon's response.

 

[ekomm]

Greatest Integer function is both differentiable and integrable 2

I appreciate the responses so far on my proposition that the greatest integer function can both be differentiated as well as integrated. Well, I want to affirm that this is true. I have done some personal exercise on this years ago just for fun and discovery. When I have time, I will upload this work. in the meantime, i want to say that it is not difficult to observe that the greatest integer function changes value only at integer boundaries -- that is -- 1, 2, 3, 4, etc. In between these numerical boundaries as I call them the greatest integer function does not change value within the intervals.

You cannot say, this function is not differentiable, so long as it changes value at the respective boundaries. Therefore, if y = [x]², its derivative, dy/dx=2[x] and not 0 as asserted by one of us. The respective derivatives at their respective boundaries are 0, 2, 4, 6, 8,10, corresponding to the ranges 0 --1, 1--2, 2-3, etc. So this function can only differentiated with respect to its respective integer boundaries and not within the respective non-integer intervals.