Evaluate $\lim_{h\to 0}\frac 1 h\int_{-\infty}^\infty g\left(\frac x h\right)f(x)\,\mathrm dx$ $g:\mathbb R\rightarrow \mathbb R$ is a continuous function such that $g(x)\geq0\, \forall x\in \mathbb R \,$ and $g(x)=0$ if $|x|\geq 1$
and $$\int_{-\infty}^\infty g(t) \,\mathbb\,dt=1$$ $f:\mathbb R\rightarrow \mathbb R$ is a continuous function. Then evaluate $$\displaystyle\lim_{h\to 0}\frac 1 h\int_{-\infty}^\infty g\left(\frac x h\right)f(x)\,\mathbb dx$$
MY TRY:By using the condition of $g$, it is clear that $$\displaystyle\int_{-1}^1 g(t) \,\mathbb\,dt=1$$But now problem is that how we implement it to solve our problem.Thank you.
Note:Ans is $f(0)$
 A: Simply substituting $u=x/h$ gives
$$ \frac{1}{h} \int_{-\infty}^{\infty} g\left(\frac{x}{h}\right)f(x)\,dx
= \int_{-\infty}^{\infty} g(u) f(hu) \,du
= \int_{-1}^{1} g(u) f(hu) \,du $$
where the latter equality follows from the definition of $g$.
Given that both $f$ and $g$ are continuous on the closed interval $[-1,1]$, they are uniformly continuous there, so we may pull the limit under the integral:
$$ \lim_{h\to 0} \frac{1}{h} \int_{-\infty}^{\infty} g\left(\frac{x}{h}\right)f(x)\,dx
= \lim_{h\to 0} \int_{-1}^{1} g(u) f(hu) \,du
=  \int_{-1}^{1} g(u) \cdot\lim_{h\to 0}f(hu) \,du$$
Finally, by continuity, $\lim_{h\to 0}f(hu) = f(0)$ and
$$ \lim_{h\to 0} \frac{1}{h} \int_{-\infty}^{\infty} g\left(\frac{x}{h}\right)f(x)\,dx 
= f(0)\int_{-1}^{1}g(u) du = f(0)$$
A: Since $g\left(x\right)=0
 $ if $\left|x\right|\geq1
 $ we have $$I_{h}=\int_{-\infty}^{\infty}g\left(\frac{x}{h}\right)f\left(x\right)dx=\int_{-h}^{h}g\left(\frac{x}{h}\right)f\left(x\right)dx
 $$ and now using the mean value theorem for integrals we have that exists some $c_{h}\in\left[-h,h\right]
 $ such that $$I_{h}=f\left(c_{h}\right)\int_{-h}^{h}g\left(\frac{x}{h}\right)dx=hf\left(c_{h}\right)\int_{-1}^{1}g\left(u\right)du=hf\left(c_{h}\right)
 $$ hence $$\lim_{h\rightarrow0}\frac{I_{h}}{h}=\lim_{h\rightarrow0}f\left(c_{h}\right)=\color{red}{f\left(0\right)}$$ from the squeeze theorem and using the fact that $f$ is continuous.
