= Solution
Use the <Euclidean norm> and its induced <operator norm>. The <spectral theorem for real symmetric matrices> gives $\|A\|=\lambda_n$ and $\|A^{-1}\|=1/\lambda_1$. If $e=y_\delta-y$, then the direct inverse has error $A^{-1}e$, so
$$
\boxed{\|A^{-1}y_\delta-x\|\leq\frac\delta{\lambda_1}.}
$$
For nonzero $y$, the relation $\|y\|=\|Ax\|\leq\lambda_n\|x\|$ also gives
$$
\boxed{\frac{\|A^{-1}y_\delta-x\|}{\|x\|}\leq\kappa\frac\delta{\|y\|},\qquad\kappa=\frac{\lambda_n}{\lambda_1}.}
$$
This is the <spectral condition number of a positive-definite matrix>. The worst relative amplification is attained by exact data along an <eigenvector> for $\lambda_n$ and noise along one for $\lambda_1$. A large $\kappa$ therefore makes the reconstruction sensitive to relative data errors. Absolute sensitivity additionally depends on $\lambda_1$ itself: scaling every <eigenvalue> equally leaves $\kappa$ unchanged but changes $1/\lambda_1$. Since the matrix is finite dimensional and positive definite, the inverse exists and is continuous; the issue is ill-conditioning rather than mathematical nonexistence or discontinuity.
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