How to prove this map is continuous? I have to prove that this matrix mapping is continuous 
  $\ H_n^+ \mapsto H_n^+ $
  $\ L \mapsto L ^ \frac{1}{2} $ 
Where $\ H_n^+ $ is the set of positive semi-definite hermitian matrices 
 A: Consider $S_n\to S$ positive-semidefinite hermitian matrices. Assume as a contradiction that $S_n^{1/2}\not\to S^{1/2}$. Id est, there is a subsequence $(S_{n_k})_{k\in\Bbb N}$ such that no subsequence of $(S_{n_k}^{1/2})_{k\in\Bbb N}$ converges to $S^{1/2}$. For each $S_{n_k}$, there is $P_{n_k}\in U(d)$ such that $P_{n_k} S_{n_k}P^H_{n_k}$ is diagonal with non-negative real entries. Since $U(d)$ is compact, we may extract a convergent subsequence $P_{n_{k_h}}\to P$. Moreover, notice that $P_{n_{k_h}}S_{n_{k_h}}P_{n_{k_h}}^H\to PSP^H$, which is diagonal with non-negative entries as well (because it's the limit of diagonal matrices with non-negative entries). Moreover, $(PSP^H)^{1/2}=PS^{1/2}P^H$ and $$(P_{n_{k_h}}S_{n_{k_h}}P_{n_{k_h}}^H)^{1/2}\to (PSP^H)^{1/2}$$
and thus $S_{n_{k_h}}^{1/2}=P_{n_{k_h}}^H(P_{n_{k_h}}S_{n_{k_h}}P_{n_{k_h}}^H)^{1/2}P_{n_{k_h}}\to P^H(PSP^H)^{1/2}P=S^{1/2}$. This is not consistent with the initial assumption that all subsequences of $(S_{n_k})_{k\in\Bbb N}$ do not converge to $S^{1/2}$.
Added: This proof can be translated without alteration to a more general context:

Let $G$ be a compact metrizable group, let $Y$ be a metric space endowed with a structure of $G$-set and let $X\subseteq Y$ be a closed subset such that $G\cdot X=Y$. Let $f:Y\to Y$ be a $G$-equivariant map such that $f(X)\subseteq X$ and $\left.f\right\rvert_X$ is continuous. Then, $f$ is continuous.

The problem at hand is the special case where $K=U(d)$, $Y=H^+_d$, $X$ is the set of diagonal matrices with non-negative real entries and $g\cdot s=gsg^H$.
