Find derivative of floor function using limit definition of derivative?

[wills921]

Homework Statement

I have been asked to find the derivative of f(x) = 0.39 + 0.24*floor(x-1) using the limit definition of a derivative. Is this possible?

Homework Equations

The Attempt at a Solution

The limit as h approaches zero of 0.24(floor(x+h-1)-floor(x-1))/h is as far as I have got. It seems the two floor functions will cancel out when substituting in 0 for h, but I'm stuck on how to get rid of the h in the denominator.

 

[Strants]

You can't substitute 0 in for h in the numerator if h is in the denominator.

Can you define a piecewise function for floor(a+b)?

 

[JonF]

Is a non-continuous function differentiable?

 

[Apphysicist]

Is a non-continuous function differentiable?

If a function is differentiable (along an interval), it is continuous (along that interval). If it is continuous, it is not necessarily differentiable.

 

[JonF]

Is this function continuous?

 

[Apphysicist]

Is this function continuous?

By definition, it has infinitely many step discontinuities.

 

[JonF]

So we are left with two options:
We can say that function isn’t differentiable as an answer.

Or we can guess that the author of the problem meant for us to look at restrictions of the domain. So for what values is this function discontinuous?

 

[Apphysicist]

So we are left with two options:
We can say that function isn’t differentiable as an answer.

Or we can guess that the author of the problem meant for us to look at restrictions of the domain. So for what values is this function discontinuous?

It is discontinuous at every integer value...floor(1.9)=1, floor(2)=2, floor(2.1)=2, floor(2.9)=2, floor(3)=3, floor(3.1)=3...

 

[JonF]

So let f_a(x) be a restriction on the domain of f(x) to (a,a+1) i.e. the continuous intervals. Can you write f_a(x) without a floor function?

 

[╔(σ_σ)╝]

JonF is correct! Adding to what he said... if a function is differentiable on an interval is has to be continuous there.
Hence, since the original function is not continuous our recourse for getting a derievative is restricting the domain to not include integers.

 

[HallsofIvy]

Let $x_0$ be a non-integer. Then there exist $\delta> 0$ such that the interval $[x_0- \delta, x_0+ \delta]$ contains only non-integers. If x and y are both in that interval, what is f(x)- f(y)?

And, of course, as other said the function is not differentiable at integer values.