If you want a more mathematical argument, then you could try something like this. let and define f by Then . When |x|>ε that gives . You can probably believe (and you may even be able to prove) that this implies that u(t)=0 whenever |x|>ε. Since ε is arbitrary, it follows that u is zero everywhere except at 0.
However, does have what is called an approximate identity. That is a sequence of elements of norm 1 such that (in the -norm) as k→∞, for all . For example, when n=1 you can take . This gives a sequence of functions that have increasingly large sharp spikes at the origin and are very small elsewhere.