# Derivative of floor function

I thought the derivative of function $\text{floor}(x)$ should be $\infty$ for integer values of $x$ and 0 elsewhere. But wolframalpha plot showed something different. Is there any explanation?

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It does not exist for integer values, and it is $0$ elsewhere. Explanation: don't trust Wolfram too much. – 1015 Feb 17 '13 at 3:58
I really appreciate this website as an atomized helping tool, and it may be a good idea to collect as much bugs we can find and raise them to the website owners to improve. – Tariq Feb 17 '13 at 4:11
I don't think being a WolframAlpha bug tracker would be a good use of this site. If you find a bug in WolframAlpha you should probably just let them know directly. – Rahul Feb 17 '13 at 4:16

The Alpha plot is badly wrong. The derivative of $\lfloor x \rfloor$ is 0 at non integers and not defined at integers. You would have to ask the people at Wolfram why this happens.

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It does that for some other graphs as well. – Joe Z. Feb 17 '13 at 4:02

It's probably calculating the derivative numerically instead of symbolically. An approach that gives a good approximation for functions that actually are differentiable, but behaves weird when the function is discontinuous.

For example, using a simple centered difference approximation $f'(x) \approx \frac{f(x + h) - f(x - h)}{2h}$, with $h = 0.05$, gives $f'(1) \approx \frac{f(1.05) - f(0.95)}{0.1} = \frac{1 - 0}{0.1} = 10$. This is nonsense, but an artifact of the computation.

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If you try asking Wolfram Alpha to differentiate the floor function, it will just output "Floor'(x)". If you force Wolfram Alpha to plot the derivative of the floor function, I think what Wolfram Alpha does is it as an infinite sum of dirac deltas, so that when you integrate, you can still get back the floor function. See http://reference.wolfram.com/mathematica/ref/DiracDelta.html and http://mathworld.wolfram.com/DeltaFunction.html

Clearly that's not what you had in mind when you asked Wolfram Alpha to plot the graph, but well...

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