Positive Semi Definite (PSD) matrix with perturbation

Question

By the definition of PSD matrix, given a vector $x \in \mathbb{R}^n$, I know that $x^T A x \geq 0$ if $A$ is PSD. I am curious that if I add a perturbation on one $x$ such that $\|x + d\| < \varepsilon$, can I establish some statement (related to $d$) that $x^T A (x + d)$ still have positive number? Any insights would be helpful.