In proof of Sard's theorem in Guillemin as well as in Milnor we consider $C$ such that if $x \in C$ then $\text{rank} \ df_x < p$ of function $f:U \rightarrow \mathbb R^p$, $U \subset R^n$ and $C_i$ such that all the partial derivatives of order $\leq i$ are $0$. In the proof of the theorem the following appears
For each $x \in C-C_1, \exists V \ \text{open}, x\in V $ such that $f(V \cap C) $ has measure $0$.
How to prove this?