Continuity of complex functions

[variety]

Do you guys know of any functions which are continuous on the real line, but discontinuous on the complex plane? If not, is there a reason why this can never happen?

 

[CompuChip]

Well, you can construct one, if you want...

Let
f(z) = \begin{cases} |z| & -1 \le \text{Im}(z)| \le 1 \\ 0 & \text{ otherwise} \end{cases}.

Why?

 

[lurflurf]

Log(1+x^2)

 

[CompuChip]

Or, inspired by lurflurf's example, 1/p(z) where p(z) is any polynomial in z without zeroes on the real line like x² + 1 or in general
p(z) = \prod_{j = 1}^N (z - a_j - b_j i)
with all the b_i not equal to zero.

And you can even multiply that by any polynomial (as long as it doesn't cancel out all the singular points of p(z)) and get another one.
And you can let N go to infinity to get a Laurent series for some function.