Hilbert transform filter phase

[russel.arnold]

We know that hilbert transform is a linear filter whose frequency response is given as -j*sgn(f), where f is the baseband frequency. Hence magnitude response of this filter is 1 and phase response is -pi/2 for f > 0 and pi/2 for f < 0. Hence phase response curve is like a staircase function ( with a shift of pi at f = 0)

Now, my doubt is can a filter whose phase response is s.t there a phase shift of pi radians at f = 0. but it increases linearly and continuously from (-inf, 0) and (0, inf) be used as a hilbert transform filter? (assuming mag response is equal to one).

 

[uart]

A linear phase shift is a constant time delay, not a constant phase delay as required.