My question is rather vague, but I will try and make it as clear as possible.
Let f be a real function of class $C^2$ on $R^2$, $r,\varepsilon>0$ two positive numbers, $x$ a critical point (i.e. such that $\nabla f(x)=0$), and $g$ "small", i.e. such that $g$ and all its partial derivatives up to order $2$ are bounded by $\varepsilon$ on $B(x,r)$.
I am looking for a quantitative condition on $f,\varepsilon,r$ that guarantees that $f+g$ has at least one critical point on $B(x,\varepsilon)$.
If for instance $x$ is a local maximum, the answer is quite easy. It is enough to assume that $$2\varepsilon<f(x)-\max_{y\in \partial B(x,r)}f(y),$$ because that condition guarantees that the maximum of $f+g$ on $B(x,r)$ is reached on the interior, therefore this local maximum will be a critical point.
But what if it is a saddle point? Is there a similar condition? I am looking for a condition valid in any dimension, but if you have an answer in dimension $2$ I am still interested. It's ok to bring asumptions on the second or third order derivatives of $f$ if needed. It's also ok to assume that $x$ is non-degenerate, i.e. that the Hessian form at $x$ is non-singular.