Sampling a Unit step function
1-) The value at t = 0 of an ideal step function converges to $\frac{f(0-) + f(0+)}{2}$ at 0 if you approach the question from a Fourier transform perspective (see Fourier transform of a discontinuous function).

2-) The analogous function in discrete time is still a unit step defined as:
$$
u[n] =
\begin{cases}
0 &amp; \text{if n &lt;  0} \\
1 &amp; \text{if n &gt;= 0} \\
\end{cases}$$

The concept of infinite frequency does not convey to discrete time representation.  In discrete time, frequency is represented by sequences of complex exponentials rather than continuous complex exponential functions.  Complex exponential sequences are not unique for frequencies above 1/2 your discrete time sampling rate so your frequency range is limited to $[0, \frac{f_s}{2}]$, where $f_s$ is your sampling rate.

See sampling theory, the nyquist frequency and a discussion of the discrete Fourier transform for a more detailed discussion.