Consider a function $f:[0,1] \rightarrow \{W \in \mathbb{R}^{n \times n}| W_{ij}=W_{ji} \}$.
As $f(t)$ for $ 0\leq t \leq 1 $ is a symmetric matrix, the function $\lambda:[0,1] \rightarrow \mathbb{R}^n$ that computes the eigenvalues of $f(t)$ and the function $v:[0,1] \rightarrow \mathbb{R}^{n \times n}$ that computes the corresponding orthogonal matrix of $f(t)$.
It is clear that if $\lambda(\cdot)$ and $v(\cdot)$ are continuous then $f(\cdot)$ is continuous.
I wonder if $f(\cdot)$ is continuous then It is necessary that $\lambda(\cdot)$ and $v(\cdot)$ are continuous?.
Thanks in advance.