Hilbert transform with strange behavior
A Hilbert Transformer is an acausal linear time-invariant (LTI) filter.  Being LTI it has all of the things an LTI filter has: the output is the convolution of the input against an impulse response.  And it has a frequency response with magnitude and phase.  

But being acausal, that means that the impulse response begins to respond before the impulse hits the input (i.e. it can see future input).  That, of course, is impossible in a real-time system so the impulse response of the Hilbert Transformer is delayed to make it causal.  That delay causes a linear trend to the phase component.

The magnitude frequency response of the Hilbert Transformer is 1 (as linear gain) or 0 dB for virtually all frequencies (DC and Nyquist are the exceptions). But without the delay, a Hilbert Transformer shifts the phase for all positive frequencies by -90° (or $-\frac{\pi}2$ radians) and all negative frequencies by +90°.  That puts a phase discontinuity at 0 Hz (DC).  Likewise, you will see the opposite phase discontinuity at Nyquist.