Discretization of Sinusoidal Signals
I have:


The function $h_1(t)=\cos(2\pi\frac14 t)$ sampled by $|||(t)$ has fourier transform $H_1(f)=4||(4f)*|||(f)=2|||(2(f-1/4))$.
The function $h_2(t)=\cos(2\pi\frac12 t)$ sampled by $|||(t)$ has fourier transform $H_2(f)=2||(2f)*|||(f)=|||(f-1/2)$,


When I bandlimit $h_1$ with $\Pi(2f)$ I obtain $4||(4f)$.
If I try to bandlimit $h_2$ by the same window, exploiting the Heaviside half maximum convention $\Pi(\pm1/2)=1/2$, I obtain $2||(2f)$.

Both sinuoids are perfectly recovered, though the Heaviside limit in the second case is not applicable in practice for dozens of reasons. 

In here, I am using the pulse, sampling and sinusoidal symbols. $H$ is the Heaviside unit step function:
$$
\Pi(f)=H(f+1/2)-H(f-1/2),\\
|||(t)=\sum_{t_n \in \mathbb{Z}}\delta(t-t_n)\\
||(t)=\frac12\delta(t+1)+\frac12\delta(t-1)
$$