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I have a function $F(x,t)=\int_0^t f(s,x)ds$ and I want to see if I can write $$\frac{\partial F(x,t)}{\partial x}=\int_0^t \frac{\partial f(s,x)}{\partial x}ds$$

So, I basically want to know if I can pass the limit of the derivative towards the inside of the integral. I am inclined to use the dominated convergence theorem but I have a basic difficulty in order to analyze if the function is pointwise convergent. How do I setup a sequence of functions $f_n(s,x)$ in order to test for convergence? My teacher said that the sequence must make sense in the context of what I am trying to find...a derivative so I am inclined to setup the sequence as $f_n(s,x+\epsilon/n)$ where $\epsilon$ is the increment in the definition of the derivative. Is this correct?

Thanks in advance.

P.S. $f(s,x)$ is a piecewise function.

The Mighty Algernon
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1 Answers1

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What I would probably try to do is look at the definition of the derivatives. (I don't know how to typeset limits properly, sorry, so ignore the line between "$lim$" and "$h \rightarrow 0$"!)

$$RHS = \int_0^t {lim \over h \rightarrow 0} [{{f(s,x+h)-f(s)}\over{h}} ds]$$ $$LHS = {lim \over h \rightarrow 0} [\int_0^t {{f(s,x+h)-f(s)}\over{h}} ds]$$

By taking sequences of functions of the functions inside the $[ \ \ ]$ brackets, then you should be able to get somewhere. Remember that if you have uniform convergence, then the integral of the limit is the limit of the integral.

Without working through the whole thing, I can't say for certain if this will work, but this should be a good starting point. :)

If this answer has helped, then please remember to upvote and/or accept this answer! :)

Sam OT
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  • thanks for your answer. My difficulty is precisely how to define the sequence a functions. For example, in a simple example, if my function is $f(s,x)=(x^{0.5}+1)e^{-s}$ I was thinking in defining the sequence as $f_n(s,x)=(x+\epsilon/n)^{0.5}+1)e^{-s}$. Does this make sense to you? – The Mighty Algernon Apr 12 '14 at 13:36
  • I don't actually know what the dominant convergence theorem is (probably just another name for something that I do know). – Sam OT Apr 12 '14 at 15:13
  • I'd try to set up some sequence and then use the fact that the uniform limit preserves integrals. With this you know that if a derivative (of cts functions $f_n$) converges uniformly, and $f$ converges somewhere, then $f$ converges uniformly, as do the derivatives, and $lim_{n \rightarrow \infty} f'_n = f'$. – Sam OT Apr 12 '14 at 15:21
  • The sequence that I would choose would then be the term in the limit of the definition of the partial derivative of $f$ wrt $x$, ie $$g_n(x,t) = \int_0^t {{f(s,x+h)-f(s)}\over{h}} ds.$$ – Sam OT Apr 12 '14 at 15:22
  • Regarding your last entry, I do not see a term $n$ in it but it seems you are saying you would relate it to $h$. Is this correct? I accepted your answer but it seems I do not have enough reputation to upvote it..sry – The Mighty Algernon Apr 12 '14 at 18:52
  • Yes, good point with the $n$: replace the $h$ by a function of $n$, eg $\epsilon / n$ (as you suggested). Remember that the limit needs to be independent of direction. (That is, $h>0$ and $h<0$ - this isn't much of a problem in real analysis, since there are only the two options (limit from above and below) - if(/when!) you get onto complex analysis, it becomes a lot harder as you can have the limit from any differentiable function (path) approaching the limit point!) – Sam OT Apr 12 '14 at 20:03
  • And with regards to the upvote, when you get enough reputation (I upvoted your question previously) I'd appreciate it if you do upvote my answer - always appreciated! :) – Sam OT Apr 12 '14 at 20:04
  • Thank you very much. I won't forget to come back here and upvote your answer when I have enough rep – The Mighty Algernon Apr 12 '14 at 20:08
  • Thank you. Hope you are able to sort this question out. :) (I've been doing analysis today also!) – Sam OT Apr 12 '14 at 20:11
  • It was very helpful thks – The Mighty Algernon Apr 12 '14 at 20:20