Example of an analytic continuation for a function in integral form

Question

Given $f(z) = \int_{-\infty}^\infty \frac{exp(-t^2)}{z-t}\,dt$, where $Im(z)>0$.

Find an analytic continuation to the region $Im(z)<0$.

Firstly the solution said that there is a branch cut on the real axis but I fail to see how. I do not see why $f(z)$ is not analytic everywhere. I considered 3 cases:

1. $Im(z)>0$

2. $z$ on real axis

3. $Im(z)<0$

Using the semicircle contour in case 1 and 3 we would find, by the residue theorem, the principal value of the integral equaling $2\pi i $ times(residue at the pole t=z). Similarly we can find for case 2, $-\pi i$ (residue at the pole t=z).

Secondly, the solution suggests that to continue $f(z)$ into the lower half plane, one should deform the contour on the real axis such that it includes the pole in lower half plane with a very sharp "spike" circulating the pole. I do not understnad this, since this way are we not just continuing to the region including only this pole?