Why the angle of a real signal in frequency is continuous?
X = angle(fft(x))


is numerically very questionable. If your signal is periodic in exactly the length of vector x you're observing, then the (theoretical) DFT would be zero in all but two bins. You can't really take the angle of $0$.

If your signal has a period that doesn't exactly fit in x, then you're observing the phase of two superimposed sinc pulses, as that's what a rectangular windowing leads to.

A sinc is for all points (but the maximum) defined as $\frac{\sin ax}{bx}$ (with $a$ and $b$ depending on who you ask and how you scale), and if you look at that, it definitely does have at least a strongly linear component.