# Continuity of complex functions

1. Feb 21, 2009

### 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?

2. Feb 22, 2009

### 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?

3. Feb 22, 2009

### lurflurf

Log(1+x^2)

4. Feb 22, 2009

### 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 bi 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.