Why are bump functions compactly supported? Smooth and compactly supported functions are called bump functions. They play an important role in mathematics and physics.
In $\mathbb{R}^n$ and $\mathbb{C}^n$, a set is compact if and only if it is closed and bounded.
It is clear why we like to work with functions that have a bounded support. But what is the advantage of working with functions that have a support that is also closed? Why do we often work with compactly supported functions, and not just functions with bounded support?

In metric spaces a real-valued function whose domain is compact is uniformly continuous and it attains a minimum and maximum value.
These properties are not to be taking lightly, for example the function $x\mapsto\frac1{x(1-x)}$ on $(0,1)$ and zero else where would be with a bounded support, but the function itself is not bounded.
Generally speaking, compact sets are very well-behaved because everything that can be characterized by open sets has, in some sense, a finite character. Continuous functions are such object, as the continuous preimage of an open set is open, so continuous functions from a compact domain have a very well-behaved nature.
The above is compatible with the following statement: Mathematicians like well-behaved objects. I have to admit that until recently I always tried to explore the naughty terrains of the mathematics and slowly I understand more and more why well-behaved objects are good. Especially when they are enough to describe a whole lot of mathematics to go around.
