Let $$\psi_{[a,b]}(x) = \begin{cases} \exp\left(-\frac{1}{(x-a)^2}-\frac{1}{(x-b)^2}\right) & \text{if }a < x < b\\0 & \text{otherwise}\end{cases}.$$
- Show that $\psi_{[a,b]}^{(n)}(x)$ exists for all $x\in \mathbb{R}$ and $n \in \mathbb{N}$.
- For each $n\in\mathbb{N}$, define $\displaystyle c_n:=\int_{-1}^1 \psi_{\left[-\frac1n,\frac1n\right]}(x)\ dx$. Show that $c_n>0$ for all $n$ but $\displaystyle \lim_{n\to\infty} c_n=0$.
- Let $f\colon [a,b]\to\mathbb{R}$ be continuous. Show that for $x_0 \in (a, b)$, $$\lim_{n\to\infty} \int_{a}^b \frac{1}{c_n} \psi_{\left[-\frac1n,\frac1n\right]}(x_0-x) f(x)\ dx = f(x_0).$$
So I know how to do #1 (likely by induction). #2 and #3 are really confusing to me but I suspect I'll have to use the fundamental theorem of calculus at some point. Can anyone help or maybe nudge me in the right direction?
Edit: I realize the first part of #2 is really easy as well. You're going to be taking the integral of an exponential function, which is always positive, so $c_n > 0$ always. I'm still stuck on how to prove that its limit is $0$ though. Is it because the integral continues to shrink (it only takes this exponential value within $a$ and $b$, thus as $n$ increases, $-1/n$ and $1/n$ become closer and closer, eventually reducing the integral to $0$?