Well, we studied that already.

We noticed, it's a continuous function as long as you stay away

from the negative real axis, on the negative real axis the argument jumps.

Because we said the argument of a number that's close to the negative real axis is

almost pi, but if you're close

to the real axis from below, then the argument is close to negative pi.

And so, as you're crossing the negative real axis,

the argument jumps from pi to -pi thereby not being continuous.

Everywhere else in the complex plane, the argument is a continuous function.

Angle that appoint forms with a positive real axis changes continuously

except the jumps over the negative real axis.

So, the argument function is continuous on the complex plane minus the negative axis

therefore, the sum of these two continuous functions is continuous

where the both are continuous.

In other words, in the complex plane minus the negative real axis.