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For the moment, consider the corresponding problem involving integration. Let $s(x)$ be the explicit solution to the following integral.

$ \displaystyle s(x)=\int_a^x f(t) \, dt $

The function $s'(x)$ is equivalent to the derivative of the integral with respect to it's upper limit and may be expressed in integral form.

$ \displaystyle s'(x)=\partial _x\left(\int_a^x f(t) \, dt\right)=f(a)+\int_a^x f'(t) \, dt $

Now let $s(x)$ be the explicit solution to the following summation.

$ \displaystyle s(x)=\sum _{t=a}^x f(t) $

The function $s'(x)$ is equivalent to the derivative of the summation with respect to it's upper limit. What is the derivative of $s(x)$ expressed in summation form?

$ \displaystyle s'(x)=\partial _x\left(\sum _{t=a}^x f(t)\right)=\ ? $

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    If $x$ is allowed to take real values, I'm expecting to see some floor functions. Else, your expression makes no sense. In the case you do introduce some floor function, you still need to see wether $s$ is differentiable or not.2012-11-30
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    Although the variable $x$ is restricted to be an integer within the summation, the derivative of $s(x)$ may still be considered without the floor function. It is assumed that $s(x)$ is differentiable. For example, if $s(x)=\frac{1}{2} x (x+1)=\sum _{t=1}^x t$, then $s'(x)=x+\frac{1}{2}=\frac{1}{2}+\sum _{t=1}^x \partial _t(t)$.2012-12-02
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    Still does not make sense to me. $\sum _{t=1}^x t$ is defined only for integer $x$. Your function on the LHS is just a real valued function that happens to coincide with the RHS for integer $x$. But that does not define $s(x)$, I could build other functions with the same property, eg: $x (x+1)/2 + sin(\pi x)$2012-12-02
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    @leonbloy, that is a fascinating difficulty you have raised. In addressing this point I think I have figured out a way to resolve the transition form x being a continuous real variable to a discrete integer variable. Define $s(x)$ by the continuous recurrence equation, $s'(x)-s'(x-1)=f'(x)$. I will post a second answer when I have completed the revision.2012-12-02
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    In addressing the difficulty leonbloy raised I realized I was implicitly using an auxiliary function, which I define in my revised answer. Now, extraneous functions of the form, $g(x) \sin (\pi x)$ get filtered out. Also, in order to avoid any misinterpretation regarding the notation used for differentiation, I would like to clarify what the [differential operator](http://en.wikipedia.org/wiki/Differential_operator) symbol $\partial _x$ means: $\partial _x\equiv D_x\equiv \frac{d}{dx}\equiv \frac{\partial }{\partial x}$.2012-12-06

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