I was reading "Discrete and continuous Fourier transforms: analysis, applications and fast algorithms" written by E.Chu and, at some point, I found something that I could not completly understand.
If $x(t)$ is not continuous, then it is possible that 2 different functions share the same transform $X(f)$. Therefore, the function $x(t)$ cannot be uniquely determined by inverse trasforming $X(f)$ unless it is continuous or it satisfies 2 more conditions:
1- $x(t)$ has only a finite number of maxima and minima on any finite interval
2- $x(t)$ has on any finite interval at most a finite number of discontinuities, each of which is a jump discontinuity.
In the latter case, the inverse Fourier transform of $X(f)$ produces $\hat{x}(t)$, which agrees with $x(t)$ at every $t$ at which $x(t)$ is continuous, and $\hat{x}(t)$ equals the average of the left-hand limit $x(t^-_{\alpha})$ and the right-hand limit $x(t^+_{\alpha})$ at every $t_{\alpha}$ at which a jump discontinuity occurs.
So, here my doubts.
First of all, I am not sure I got the meaning of the statements related to the 2 conditions listed. This is probably due to the fact that I am familiar only with signals coming from structural tests (mainly accelerations), therefore I think it could be really helpful if someone of you can provide me some example I can focus on to understand the concept (for example, how can be possible to have infinite maxima and minima in a signal? And what are the discontinuities it is talking about?).
Second, measured signals (for example, the aformentioned acceleration measurements coming from a test) are not continuous (since they are sampled at a given sampling frequency), so if I want to apply the IFFT algorithm to retrieve the time signal does it mean I have always to check if the 2 other conditions are satisfied? And how can I check them? Visual inspection? Or do I have to look for some specific features the signal must have?
Are there classes of signals which always satisfy the 2 conditions?