When I was doing revision for definite integral, I got confused with one step.
Here is the statement:
For all continuous functions $f$ which are continuous at $0$:
$$\lim_{ε\to 0}\int_{-ε}^ε f(x)·D(x,ε)dx = f(0)·\lim_{ε\to 0}\int_{-ε}^ε D(x,ε)dx$$
where $D(x,ε)$is the Dirac Delta function.
But I don't really understand why is this true. Could anyone tell me the steps between them?
Rewritten as a teaching exercise:
Please explain why for all functions $f$ continuous at $0$:
$$\lim\limits_{\epsilon \to 0} \int\limits_{-\epsilon}^\epsilon f(x) \delta(x)\ dx = f(0) \lim_{\epsilon \to 0}\int\limits_{-\epsilon}^\epsilon \delta(x)\ dx = f(0).$$
where $\delta$ is a Dirac delta function.
Here is the definition of Dirac delta function
$$\lim_{\epsilon \to 0}\delta(t;\epsilon)=0\quad\forall t\neq0$$ and $$\lim_{\epsilon \to 0}\int_{-\infty}^\infty \delta(t;\epsilon)dt=1$$
