A delta function is not a function in the normal sense. Infinity is not a proper number.
However, delta functions have a property that can be used to
define them. That property is called the “filtering property.” If you multiply a delta function by a
smooth function and integrate over all , you get the value
of the function at the location of the delta function:
You can reverse that statement and define the delta function as the “distribution” that produces the result above for any smooth function . (The functions are normally further constrained by a requirement that they must become zero at their ends.)
In a similar way you can also define the derivative of the delta
function, the dipole . It is the distribution for which,
for any smooth ,