Let $f:\{-1,1\}^n\to\{-1,1\}$ be a boolean function. Define the influence of the $i$'th coordinate of $f$ as follows: $\operatorname{Inf}_i(f)=\Pr_{x}[f(x)\neq f(\hat x_i)]$ where $x$ is uniformly picked from $\{-1,1\}^n$, and $\hat x_i$ is $x$ with its $i$'th coordinate flipped (e.g., say $x=(1,1,1,1,-1)$, then $\hat x_3=(1,1,-1,1,-1)$).
How do I show that if the influence of a boolean function is $1$ at each coordinate, then the boolean function must be the $\pm$parity function ?
Formally speaking, if a boolean function $f$ holds $\displaystyle\sum_{i=1}^n \operatorname{Inf}_i(f)=n$ then $\displaystyle f(x_1,\ldots,x_n)=\pm\prod_{i=1}^n x_i$