Let $0\le a < b \le1$ and $\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$. I want to know if the following statement is true: $$ \sum_{n\in \mathbb Z} \int_{n+a}^{n+b} \phi(x) dx = b-a, $$ where the summation runs over all integer numbers.
It is correct if $a = 0, b=1$, since $\phi$ is the density of the standard normal distribution. It is also correct if $a = 0, b= 1/2$ by some arguments with symmetry. Is it correct for arbitrary choice of $a$ and $b$? Is there elementary approach to prove it?
