Prove $\lim_{y \to 0^+} \frac{y}{\pi} \int_{-\infty}^{+\infty} \frac{f(x) \, \mathrm{d}x}{x^2+y^2} = f(0)$ with $f$ continuous, bounded

Question

I have been trying to solve the following problem, given $f$ continuous and bounded prove that

$$\lim_{y \to 0^+} \frac{y}{\pi} \int_{-\infty}^{+\infty} \frac{f(x) \, \mathrm{d}x}{x^2+y^2} = f(0)$$

I tried first solving this by parts by integrating $\frac{1}{x^2+y^2}$ and differentiating $f(x)$ but this doesn't seem to work or bring me anywhere.

Answer 1

First note that for any $y>0$

$$\int_{-\infty}^\infty \frac{y}{\pi(x^2+y^2)}\,dx=1$$

So, it suffices to show that

$$\lim_{y\to 0^+}\int_{-\infty}^\infty \frac{y(f(x)-f(0))}{\pi(x^2+y^2)}\,dx=0$$

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Given $\epsilon>0$, fix a number $\delta>0$ such that $|f(x)-f(0)|<\epsilon$ for $|x|<\delta$.

With this fixed $\delta$, we write for $y>0$

$$\begin{align} \left|\int_{-\infty}^\infty \frac{y(f(x)-f(0))}{\pi(x^2+y^2)}\,dx\right|&\le\left|\int_{|x|\le \delta} \frac{y(f(x)-f(0))}{\pi(x^2+y^2)}\,dx\right|+\left|\int_{|x|\ge\delta} \frac{y(f(x)-f(0))}{\pi(x^2+y^2)}\,dx\right|\\\\ &\le \int_{|x|\le \delta} \frac{y\left|f(x)-f(0)\right|}{\pi(x^2+y^2)}\,dx+ \int_{|x|\ge \delta} \frac{y\left|f(x)-f(0)\right|}{\pi(x^2+y^2)}\,dx\\\\ &\le \frac{2\epsilon}\pi \arctan(\delta/y)+\frac{2\sup_{|x|\ge\delta }(f(x))}\pi \left(\pi/2-\arctan(\delta/y)\right)\tag 1 \end{align}$$

Letting $y\to 0^+$ in $(1)$, we find that for any given $\epsilon>0$,

$$\lim_{y\to 0^+}\left|\int_{-\infty}^\infty \frac{y(f(x)-f(0))}{\pi(x^2+y^2)}\,dx\right|\le \epsilon$$

And we are done!

Answer 2

$${y\over\pi}\int_{-\infty}^\infty {f(x)\, dx \over x^2 + y^2} = {y\over\pi}\int_{-\infty}^\infty {f(xy)y\,dx\over y^2x^2 + y^2 } = {1\over \pi}\int_{-\infty}^\infty {f(xy)\,dx\over 1 + x^2}.$$ Now apply bounded convergence against the measure $dx/(1+x^2)$.