Mathematical conditions for the existence of the Fourier transform can be
very involved. The following is a working definition:
A sufficient, but not necessary, condition for the existence of the Fourier
Transform 
of 
is that 
is absolutely integrable
function, i.e.,
and that it has only a finite number of finite discontinuities. This requirement is generally met for real-world signals. However, this requirement is not met for Euler functions or for periodic functions which, strictly speaking, do not have Fourier Transforms. Instead, we must use the concept of the transform in the limit. If such a limit exists, we can use the limit as the ``function'' tranform. This convenience is very useful in engineering applications.