1) Why is true that there is an open ball in $M_n$ centered at the identity matrix and a continuous function $f$ defined on this open ball s.t. $f(M)^2=M$ for $M$ in the ball?
2) Extending the question a bit: Would I be right in thinking that we cannot do the same for any arbitrary matrix $M\in M_n$ because considering $n=1$ this clearly fails for the negative numbers?
3) What about higher powers? The case $n=1$ clearly works for all odd powers. But not for the even ones. But perhaps there is a catch for the odd powers too?
Thanks.