If there is a limit of $u(r,t)$ when $r\to 0$, we can always define $u(0,t)$ so that it will be continuous and, as long we deal with ordinary functions rather than distribution, we always fix functions this way. We call this *a removable singularity*.

For ordinary functions not only single points but the *sets of measure zero *(see Real Analysis class, in our class we will talk about this much later) do not matter.

For *distributions* it is not so, as they could be *supported* at single points.

Now, to finish this problem just write down $u$, plugging correct $g=-f$