Dirac delta function integral only at a specific point

Question

From what I understand the Dirac delta function is infinitely thin at every point except for one, where its infinite.
I was wondering if we completely disregard the function at every point except for where its infinite,
then is it possible to rewrite it as : $$\int_{- \infty}^{+\infty} \delta (x) = \lim_{\epsilon \rightarrow 0} \int_{- \epsilon}^{+\epsilon} \delta (x) = 1 $$ or in a more general case : $$ \int_{- \infty}^{+\infty} \delta (x - a) \psi (x) = \lim_{\epsilon \rightarrow 0} \int_{- \epsilon}^{+\epsilon} \delta (x-a) \psi(x) = \psi(a) $$ Any help would be appreciated.