Fourier phase (unwrapping problem)

[mnb96]

Hello,
given an integrable function f(x), and its Fourier transform

$$\mathcal{F}\{f\}(\omega)=\int_{\mathbb{R}}f(x)e^{-i\omega x}dx$$,

we consider the phase $$\mathrm{Ph}_f : \mathbb{R}\rightarrow [-\pi,\pi)$$ which is given by:

$$\mathrm{Ph}_f (\omega) = \mathrm{arg}(\mathcal{F}\{f\}(\omega))$$

In general the phase function will have discontinuities (when it wraps from $-\pi$ to $\pi$, and there are algorithms that attempts to recover a continuous phase function.
My question is: why should the phase be a continuous function? What is the condition/theorem that guarantees that the phase is always continuous?

 

[Xitami]

"[URL Scientist and Engineer's Guide to Digital Signal Processing
By Steven W. Smith, Ph.D.[/URL]

• 
  http://www.dspguide.com/ch8.htm"
  • http://www.dspguide.com/ch8/8.htm"
    
  • http://www.dspguide.com/ch8/9.htm"

 

[mnb96]

I am afraid that chapter doesn't answer my question which was:

"...Is there a condition on f (or a theorem) that guarantees that the phase is certainly continuous? ..."