Non degenerate critical points.

Question

Let $K$ be a compact subset of the Euclidean space $\mathbb{R}^n$; let $U$ be a neighborhood of $K$ and let $f:U \rightarrow \mathbb{R}$ be a smooth function such that all critical points of $f$ in $K$ have index $\ge \lambda_0$. If $g:U \rightarrow \mathbb{R}$ is a smooth funtion such that $$ \left|\frac{\partial g}{\partial x_{i}} - \frac{\partial f}{\partial x_{i}} \right|< \epsilon $$ and $$ \left|\frac{\partial^{2}g}{\partial x_{i} \partial x_{j}}- \frac{\partial^{2}f}{\partial x_{i} \partial x_{j}}\right|< \epsilon ,$$ for all $i,j \in [1,n]$ uniformly throughout $K$, for sufficiente small costant $\epsilon$, then alla critical points of $g$ in $K$ have index $\ge \lambda_0$. Can you give me an example of functions $f$ and $g$ with these proprerties? (for example if $n=1,2$)