Show that there is no continuous square root function defined on all of \(\CC\).
Solution
Suppose \(f(z)^2 = z\). Then \(f\) is a section to the covering map \[ p: \unitsof{\CC} &\to \unitsof{\CC} \\ z & \mapsto z^2 ,\] so \(p\circ f = \id\). Using \(\pi_1(\unitsof{\CC}) = \ZZ\), the induced maps are \(p_*(1) = 2\) and \(f_*(1) = n\) for some \(n\in \ZZ\). But then \(p_* \circ f_*\) is multiplication by \(2n\), contradicting \(p_* \circ f_* = \id\) by functoriality.