Discontinuity in the angle of a complex exponential signal Plotting this complex expontential:

$$z[n]=100e^{j0.1n},\quad\text{we have a discontinuity at}\quad \arg[n]=10\pi$$



I guess this is related to the angle formula:

$$\theta=\arctan\left(\frac{\Im\{z[n]\}}{\Re\{z[n]\}}\right)$$

But I do not quite see the relationship. Can anyone tell me exactly why we have a discontinuity at $10\pi$?
That's not really a discontinuity. On a circle the two points $-\pi$ and $+\pi$ are identified: They are the same point. That is true for all $x$ and $x+n 2\pi$ for integer $n$.

If you would like to keep the continuity obvious, you might want to switch to a phasor representation of angles. Those are complex numbers on the unit circle $\exp(i\phi)$ and they form a group under multiplication. They nicely absorb the non-obvious topology of the phase angle.