I know that there are questions on this website on how to prove that the integral of the Dirac function is $1$. However, I am interested in knowing why my proof is wrong and if there is a way to salvage it. So here's the question :
The Dirac delta function can be defined by the limit of a short pulse: $$\delta(t-t_0) = \lim_{\Delta \rightarrow 0}({f_{\Delta}(t)})$$ where $f_{\Delta}(t) = \frac{1}{\Delta}$ for $t_0\leq t\leq t_0 + \Delta$ and $f_{\Delta}(t) = 0$ otherwise. Convince yourself that the integral $\int_{t_1}^{t_2} \delta(t-t_0) dt = 1$ if $t_1\leq t_0 \leq t_2 $
Here's my approach:
$$\int_{t_1}^{t_2} \delta(t-t_0) dt = \int_{t_1}^{t_2} \lim_{\Delta \rightarrow 0}({f_{\Delta}(t)}) dt$$ $$=\int_{t_1}^{t_2} \lim_{\Delta \rightarrow 0}(\frac{1}{\Delta})dt$$ $$=\lim_{\Delta \rightarrow 0}(\frac{1}{\Delta})\int_{t_1}^{t_2}dt$$ $$=\lim_{\Delta \rightarrow 0}(\frac{1}{\Delta}) \cdot (t_2-t_1)$$ $$=\lim_{\Delta \rightarrow 0}(\frac{t_2-t_1}{\Delta})$$ If we let $t_1 = t_0$ and $t_2 = t_0 + \Delta$, then we get $$\lim_{\Delta \rightarrow 0}(\frac{\Delta}{\Delta})$$ $$ = 1$$
Here's why I doubt my proof.
First, in the third line of my development, I took out the limit from the integral. My reasoning behind the maneuver is that this limit does not depend on $t$ and therefore I can treat it as a constant. However, upon re-reading my proof, I am wondering if I am really "allowed" to do this. Since I am treating the limit as a constant, I am assuming that it exists and is finite, which is not the case, right? (since $\lim_{x \rightarrow 0} (1/x)$ does not exist).
Secondly, to get to the 6th line, I defined $t_2 = t_0 + \Delta$. Am I allowed to do this? I defined it that way only because it was mathematically convenient, not because it actually makes sense, which is why I doubt its validity.
I personally think that the proof is incorrect, but I'm not sure. Perhaps one could argue otherwise and I would be interested to know how.
So, can you confirm to me that my proof is incorrect, and can you please let me know if there's anything I can do to "salvage" it, or if I need to try another approach
Best Answer
Your problem begins with the definition of Dirac delta $$ \delta(t-t_0) = \lim_{\Delta \rightarrow 0}({f_{\Delta}(t)}) \tag{1} $$ which is not an equality of functions since the pointwise limit does not exist at $\,t=t_0.\,$ Thus, your very first step of writing $$\int_{t_1}^{t_2} \delta(t-t_0) dt = \int_{t_1}^{t_2} \lim_{\Delta \rightarrow 0}({f_{\Delta}(t)}) dt \tag{2} $$ is problematic unless interpreted as an appropriate kind of integral suitable for distributions. More problematic is that, as the Wikipedia article Dirac delta function states:
Next, your step equating the integral to $$ =\lim_{\Delta \rightarrow 0}(\frac{1}{\Delta})\int_{t_1}^{t_2}dt \tag{3} $$ is flawed because you should not be integrating the constant function $1$ but instead the function $\,\Delta f_\Delta(t).\,$ As the definition states:
What you are integrating here has the value $1$ for $t_0\leq t\leq t_0 + \Delta$ and $0$ otherwise. Thus the integral is not $\,(t_2-t_1)\,$ as you have it in your next step.