Is this equality correct?

Question

I am working on a problem and stuck at some point. By intuition I believe that the equality below should hold. Then the bigger problem makes sense. However, I could not prove it. Does anybody prove or disprove the below equality?

$$ \frac{\partial u(\operatorname{floor}(x))}{\partial u(x)} \> \xi(x) \stackrel{?}{=} \xi(\operatorname{floor}(x)) $$

Thanks in advance.

Answer 1

Do you mean the functional derivative (usually written $\delta$, not $\partial$)? Then your equality is incorrect.

Let $y = \text{floor}(x)$. If $x$ is not an integer, $x \ne y$, and then $\dfrac{\delta u(y)}{\delta u(x)} = 0$.