Convolution of Real Function with Rectangle Function

Theorem
Let $f: \R \to \R$ be a real function.

Consider the rectangle function $\Pi: \R \to \R$.

Then:
 * $\forall x \in \R: \map \Pi x * \map f x = \displaystyle \int_{x \mathop - \frac 1 2}^{x \mathop + \frac 1 2} \map f u \rd u$

where $*$ denotes the convolution integral.